HOMEWORK ASSIGNMENT 9
ACCELERATED PROOFS AND PROBLEM SOLVING [MATH08071]
Each problem will be marked out of 4 points.
Exercise 1 ([1, Exercise 20.2]). Let f and g be the following permutations in S7 :
1 2 3 4 5 6 7
1 2 3 4 5 6 7
f=
, g=
.
3 1 5 7 2 6 4
3 1 7 6 4 5 2
Write down in cycle notation the permutations f , g, g 2 , g 3 , f ◦ g, (f ◦ g)−1 , g −1 ◦ f −1 .
What is the order of f ? What is the order of f ◦ g?
Solution. We have f = (1 3 5 2)(4 7), g = (1 3 7 2)(4 6 5), g 2 = (1 7)(3 2)(4 5 6),
g 3 = (1 2 7 3), f ◦ g = (1 5 7)(2 3 4 6), (f ◦ g)−1 = (1 7 5)(2 6 4 3), and
g −1 ◦ f −1 = (f ◦ g)−1 = (1 7 5)(2 6 4 3).
The order of f is 4 by [1, Proposition 20.4]. Similarly, the order of f ◦ g is 12.
Exercise 2 ([1, Exercise 20.4]). A pack of 2n cards is shuffled by the “interlacing” method
described in [1, Example 20.7] — in other words, if the original order is 1, 2, 3, 4, . . . , 2n,
the new order after the shuffle is 1, n + 1, 2, n + 2, . . . , n, 2n. Work out how many times this
shuffle must be repeated before the cards are again in the original order in the following
cases:
(a) n = 10
(b) n = 12
(c) n = 14
(d) n = 16
(e) n = 24
(f) n = 26 (i.e., a real pack of cards).
Investigate this question as far as you can for a general n — it is quite fascinating.
Solution. In other words, we must find the order of the permutation
1
2
3
4
5
6
· · · 2n − 1 2n
∈ S2n
1 n + 1 2 n + 2 3 n + 3 ···
n
2n
for n = 10, 12, 14, 16, 24, 26. Let us find the order of the above permutation in the cases
n = 10, 12, 14, 16, 24, 26 step by step. See Appendix for further investigation (for all n
from 4 till 1000).
(10) The case n = 10. The first cycle is a cycle of length 18. This cycle looks like this:
(2 11 6 13 7 4 12 16 18 19 10 15 8 14 17 9 5 3).
The permutation is a composition of 3 cycles of lengths 1, 18, and 1. Namely, we
have the following composition of cycles:
(1)(2 11 6 13 7 4 12 16 18 19 10 15 8 14 17 9 5 3)(20).
Omitting the cycles of length one, we have the following composition of cycles:
(2 11 6 13 7 4 12 16 18 19 10 15 8 14 17 9 5 3).
This assignment is due on Thursday 26th November 2015.
1
So, the order of the permutation is 18 by [1, Proposition 20.4].
(12) The case n = 12. The first cycle is a cycle of length 11. This cycle looks like this:
(2 13 7 4 14 19 10 17 9 5 3).
The permutation is a composition of 4 cycles of lengths 1, 11, 11, and 1. Namely,
we have the following composition of cycles:
(1)(2 13 7 4 14 19 10 17 9 5 3)(6 15 8 16 20 22 23 12 18 21 11)(24).
Omitting the cycles of length one, we have the following composition of cycles:
(2 13 7 4 14 19 10 17 9 5 3)(6 15 8 16 20 22 23 12 18 21 11).
So, the order of the permutation is 11 by [1, Proposition 20.4].
(14) The case n = 14. The first cycle is a cycle of length 18. This cycle looks like this:
(2 15 8 18 23 12 20 24 26 27 14 21 11 6 17 9 5 3).
The permutation is a composition of 5 cycles of lengths 1, 18, 6, 2, and 1. Namely,
we have the following composition of cycles:
(1)(2 15 8 18 23 12 20 24 26 27 14 21 11 6 17 9 5 3)(4 16 22 25 13 7)(10 19)(28).
Omitting the cycles of length one, we have the following composition of cycles:
(2 15 8 18 23 12 20 24 26 27 14 21 11 6 17 9 5 3)(4 16 22 25 13 7)(10 19).
So, the order of the permutation is 18 by [1, Proposition 20.4].
(16) The case n = 16. The first cycle is a cycle of length 5. This cycle looks like this:
(2 17 9 5 3).
The permutation is a composition of 8 cycles of lengths 1, 5, 5, 5, 5, 5, 5, and 1.
Namely, we have the following composition of cycles:
(1)(2 17 9 5 3)(4 18 25 13 7)(6 19 10 21 11)(8 20 26 29 15)(12 22 27 14 23)(16 24 28 30 31)(32).
Omitting the cycles of length one, we have the following composition of cycles:
(2 17 9 5 3)(4 18 25 13 7)(6 19 10 21 11)(8 20 26 29 15)(12 22 27 14 23)(16 24 28 30 31).
So, the order of the permutation is 5 by [1, Proposition 20.4].
(24) The case n = 24. The first cycle is a cycle of length 23. This cycle looks like this:
(2 25 13 7 4 26 37 19 10 29 15 8 28 38 43 22 35 18 33 17 9 5 3).
The permutation is a composition of 4 cycles of lengths 1, 23, 23, and 1. So, the
order of the permutation is 23 by [1, Proposition 20.4].
(26) The case n = 26. The first cycle is a cycle of length 8. This cycle looks like this:
(2 27 14 33 17 9 5 3).
The permutation is a composition of 9 cycles of lengths 1, 8, 8, 8, 8, 8, 2, 8, and
1. Omitting the cycles of length one, we have the following composition of cycles:
(2 27 14 33 17 9 5 3)(4 28 40 46 49 25 13 7)(6 29 15 8 30 41 21 11)
(10 31 16 34 43 22 37 19)(12 32 42 47 24 38 45 23)(18 35)(20 36 44 48 50 51 26 39).
So, the order of the permutation is 8 by [1, Proposition 20.4]. Note that 8 is
relatively small comparing to the pack size 52. I guess this is the reason why this
interlacing trick is so popular among gamblers.
2
Exercise 3 ([1, Exercise 20.6]). This question is about the 3 × 3 version of the Fifteen
Pizzle of Example 20.10. Starting with the configuration
1 2 3
4 5 6
7 8 which of the following configurations can be reached by a sequence of moves?
3 2 1
4 5 6,
7 8 1 2
3 4 5,
6 7 8
Solution. Arguing as in [1, Example 20.10],
ration
3 2
4 5
7 8
1 7 2
6 4 5
3 8 we see that we never can reach the configu1
6,
because the corresponding permutation
1 2 3 4 5 6 7 8
= (1 3)
3 2 1 4 5 6 4 8
is odd. On the other hand, we can reach the remaining two configurations
1 2
3 4 5
6 7 8
and
1 7 2
6 4 5.
3 8 Let us show this other way around. Namely, let us show how to start with any of this
two configuration and get the initial configuration
1 2 3
4 5 6.
7 8 For the first case we have
1 2
1 4 2
1 4 2
1 4 2
1 4 2
1 4 2
3 4 5 −→ 3 5 −→ 3 5 −→ 6 3 5 −→ 6 3 5 −→ 6 3 5 −→
6 7 8
6 7 8
6 7 8
7 8
7 8
7 8 1 4 2
1 4 2
1 4 2
1 4 2
1 4 2
1 4 2
−→ 6 3 −→ 6 3 −→ 6 3 −→ 7 6 3 −→ 7 6 3 −→ 7 3 −→
7 8 5
7 8 5
7 8 5
8 5
8 5
8 6 5
1 2
1 2 1 2 3
1 2 3
1 2 3
1 2 3
−→ 7 4 3 −→ 7 4 3 −→ 7 4 −→ 7 4 5 −→ 7 4 5 −→ 7 4 5 −→
8 6 5
8 6 5
8 6 5
8 6 8 6
8 6
1 2 3
1 2 3
1 2 3
1 2 3
−→ 4 5 −→ 4 5 −→ 4 5 −→ 4 5 6 .
7 8 6
7 8 6
7 8 6
7 8 For the second case we have
1 7 2
1 7 2
1 7 2
1 7 2
1 7 2
6 4 5 −→ 6 4 5 −→ 6 4 5 −→ 4 5 −→ 4 5 −→
3 8 3 8
3 8
6 3 8
6 3 8
3
1 7 2
1 7 2
1 7 2
1 7 2
1 7 2
−→ 4 3 5 −→ 4 3 5 −→ 4 3 −→ 4 3 −→ 4 8 3 −→
6 8
6 8 6 8 5
6 8 5
6 5
1 7 2
1 7 2
1 7 2
1 2
1 2 −→ 4 8 3 −→ 8 3 −→ 8 3 −→ 8 7 3 −→ 8 7 3 −→
6 5
4 6 5
4 6 5
4 6 5
4 6 5
1 2 3
1 2 3
1 2 3
1 2 3
1 2 3
−→ 8 7 −→ 8 7 5 −→ 8 7 5 −→ 8 5 −→ 8 5 −→
4 6 5
4 6 4 6
4 7 6
4 7 6
1 2 3
1 2 3
1 2 3
1 2 3
1 2 3
−→ 4 8 5 −→ 4 8 5 −→ 4 5 −→ 4 5 −→ 4 5 6 .
7 6
7 6
7 8 6
7 8 6
7 8 Now reversing the arrows, we start with the configuration
1 2 3
4 5 6
7 8 and get the following configurations
1 2
3 4 5
6 7 8
and
1 7 2
6 4 5.
3 8 Exercise 4 ([3, Problem 5]). Denote by α the permutation
1 2 3 4 5 6 7
∈ S7 .
3 7 4 1 6 5 2
(a) Express α as a product of disjoint cycles.
(b) Find the order of the permutation α and find
|α ◦ α ◦ α ◦{z· · · ◦ α ◦ α} .
2011 times
Solution. α = (1 3 4)(2 7)(5 6). Then the order of α is 6 by [1, Proposition 20.4]. Then
335
2011
335×6+1
α
◦
α
◦
α
◦
·
·
·
◦
α
◦
α
=
α
=
α
=
α6
α = α = (1 3 4)(2 7)(5 6).
|
{z
}
2011 times
Exercise 5 ([2, Problem 5]). Let f, g be the permutation in S7 .
1 2 3 4 5 6 7
1 2 3 4 5 6 7
f=
, g=
.
4 5 7 3 2 6 1
7 6 4 5 3 2 1
(i)
(ii)
(iii)
(iv)
Calculate f −1 and f ◦ g.
Express f as a composite of disjoint cycles.
Determine whether f is an even or odd permutation.
Write g as a composite of simple interchanges.
Solution. We have
1 2 3 4 5 6 7
1 2 3 4 5 6 7
−1
f =
and f ◦ g =
.
7 5 4 1 2 6 3
1 6 3 2 7 5 4
We have f = (1 4 3 7)(2 5). Then f is even by [1, Proposition 20.7].
4
Recall that simple interchange is just another name for a cycle of length 2. Thus,
arguing as in the proof of [1, Proposition 20.6], we see that
1 2 3 4 5 6 7
g=
= (1 7)(2 6)(3 4 5) = (1 7)(2 6)(3 5)(3 4).
7 6 4 5 3 2 1
Appendix A. Interlacing cards
Let us use assumptions and notations of [1, Problem 20.4]. Let us first describe cycle
decomposition for the permutation in [1, Problem 20.4] for all n from 4 to 19.
(4) The case n = 4. The permutation looks like this:
1 2 3 4 5 6 7 8
1 5 2 6 3 7 4 8
The first cycle is a cycle of length 3. This cycle looks like this:
(2 5 3).
The permutation is a composition of 4 cycles of lengths 1, 3, 3, and 1. Namely, we
have the following composition of cycles:
(1)(2 5 3)(4 6 7)(8).
Omitting the cycles of length one, we have the following composition of cycles:
(2 5 3)(4 6 7).
So, the order of the permutation is 3.
(5) The case n = 5. The permutation looks like this:
1 2 3 4 5 6 7 8 9 10
1 6 2 7 3 8 4 9 5 10
The first cycle is a cycle of length 6. This cycle looks like this:
(2 6 8 9 5 3).
The permutation is a composition of 4 cycles of lengths 1, 6, 2, and 1. Namely, we
have the following composition of cycles:
(1)(2 6 8 9 5 3)(4 7)(10).
Omitting the cycles of length one, we have the following composition of cycles:
(2 6 8 9 5 3)(4 7).
So, the order of the permutation is 6.
(6) The case n = 6. The permutation looks like this:
1 2 3 4 5 6 7 8 9 10 11 12
1 7 2 8 3 9 4 10 5 11 6 12
The first cycle is a cycle of length 10. This cycle looks like this:
(2 7 4 8 10 11 6 9 5 3).
The permutation is a composition of 3 cycles of lengths 1, 10, and 1. Namely, we
have the following composition of cycles:
(1)(2 7 4 8 10 11 6 9 5 3)(12).
Omitting the cycles of length one, we have the following composition of cycles:
(2 7 4 8 10 11 6 9 5 3).
So, the order of the permutation is 10.
5
(7) The case n = 7. The permutation looks like this:
1 2 3 4 5 6 7 8 9 10 11 12 13 14
1 8 2 9 3 10 4 11 5 12 6 13 7 14
The first cycle is a cycle of length 12. This cycle looks like this:
(2 8 11 6 10 12 13 7 4 9 5 3).
The permutation is a composition of 3 cycles of lengths 1, 12, and 1. Namely, we
have the following composition of cycles:
(1)(2 8 11 6 10 12 13 7 4 9 5 3)(14).
Omitting the cycles of length one, we have the following composition of cycles:
(2 8 11 6 10 12 13 7 4 9 5 3).
So, the order of the permutation is 12.
(8) The case n = 8. The permutation looks like this:
1 2 3 4 5 6 7 8 9 10 11 12 13 14 15 16
1 9 2 10 3 11 4 12 5 13 6 14 7 15 8 16
The first cycle is a cycle of length 4. This cycle looks like this:
(2 9 5 3).
The permutation is a composition of 6 cycles of lengths 1, 4, 4, 2, 4, and 1. Namely,
we have the following composition of cycles:
(1)(2 9 5 3)(4 10 13 7)(6 11)(8 12 14 15)(16).
Omitting the cycles of length one, we have the following composition of cycles:
(2 9 5 3)(4 10 13 7)(6 11)(8 12 14 15).
So, the order of the permutation is 4.
(9) The case n = 9. The first cycle is a cycle of length 8. This cycle looks like this:
(2 10 14 16 17 9 5 3).
The permutation is a composition of 4 cycles of lengths 1, 8, 8, and 1. Namely, we
have the following composition of cycles:
(1)(2 10 14 16 17 9 5 3)(4 11 6 12 15 8 13 7)(18).
Omitting the cycles of length one, we have the following composition of cycles:
(2 10 14 16 17 9 5 3)(4 11 6 12 15 8 13 7).
So, the order of the permutation is 8.
(10) The case n = 10. The first cycle is a cycle of length 18. This cycle looks like this:
(2 11 6 13 7 4 12 16 18 19 10 15 8 14 17 9 5 3).
The permutation is a composition of 3 cycles of lengths 1, 18, and 1. Namely, we
have the following composition of cycles:
(1)(2 11 6 13 7 4 12 16 18 19 10 15 8 14 17 9 5 3)(20).
Omitting the cycles of length one, we have the following composition of cycles:
(2 11 6 13 7 4 12 16 18 19 10 15 8 14 17 9 5 3).
So, the order of the permutation is 18.
6
(11) The case n = 11. The first cycle is a cycle of length 6. This cycle looks like this:
(2 12 17 9 5 3).
The permutation is a composition of 7 cycles of lengths 1, 6, 3, 6, 2, 3, and 1.
Namely, we have the following composition of cycles:
(1)(2 12 17 9 5 3)(4 13 7)(6 14 18 20 21 11)(8 15)(10 16 19)(22).
Omitting the cycles of length one, we have the following composition of cycles:
(2 12 17 9 5 3)(4 13 7)(6 14 18 20 21 11)(8 15)(10 16 19).
So, the order of the permutation is 6.
(12) The case n = 12. The first cycle is a cycle of length 11. This cycle looks like this:
(2 13 7 4 14 19 10 17 9 5 3).
The permutation is a composition of 4 cycles of lengths 1, 11, 11, and 1. Namely,
we have the following composition of cycles:
(1)(2 13 7 4 14 19 10 17 9 5 3)(6 15 8 16 20 22 23 12 18 21 11)(24).
Omitting the cycles of length one, we have the following composition of cycles:
(2 13 7 4 14 19 10 17 9 5 3)(6 15 8 16 20 22 23 12 18 21 11).
So, the order of the permutation is 11.
(13) The case n = 13. The first cycle is a cycle of length 20. This cycle looks like this:
(2 14 20 23 12 19 10 18 22 24 25 13 7 4 15 8 17 9 5 3).
The permutation is a composition of 4 cycles of lengths 1, 20, 4, and 1. Namely,
we have the following composition of cycles:
(1)(2 14 20 23 12 19 10 18 22 24 25 13 7 4 15 8 17 9 5 3)(6 16 21 11)(26).
Omitting the cycles of length one, we have the following composition of cycles:
(2 14 20 23 12 19 10 18 22 24 25 13 7 4 15 8 17 9 5 3)(6 16 21 11).
So, the order of the permutation is 20.
(14) The case n = 14. The first cycle is a cycle of length 18. This cycle looks like this:
(2 15 8 18 23 12 20 24 26 27 14 21 11 6 17 9 5 3).
The permutation is a composition of 5 cycles of lengths 1, 18, 6, 2, and 1. Namely,
we have the following composition of cycles:
(1)(2 15 8 18 23 12 20 24 26 27 14 21 11 6 17 9 5 3)(4 16 22 25 13 7)(10 19)(28).
Omitting the cycles of length one, we have the following composition of cycles:
(2 15 8 18 23 12 20 24 26 27 14 21 11 6 17 9 5 3)(4 16 22 25 13 7)(10 19).
So, the order of the permutation is 18.
(15) The case n = 15. The first cycle is a cycle of length 28. This cycle looks like this:
(2 16 23 12 21 11 6 18 24 27 14 22 26 28 29 15 8 19 10 20 25 13 7 4 17 9 5 3).
The permutation is a composition of 3 cycles of lengths 1, 28, and 1. Namely, we
have the following composition of cycles:
(1)(2 16 23 12 21 11 6 18 24 27 14 22 26 28 29 15 8 19 10 20 25 13 7 4 17 9 5 3)(30).
Omitting the cycles of length one, we have the following composition of cycles:
(2 16 23 12 21 11 6 18 24 27 14 22 26 28 29 15 8 19 10 20 25 13 7 4 17 9 5 3).
So, the order of the permutation is 28.
7
(16) The case n = 16. The first cycle is a cycle of length 5. This cycle looks like this:
(2 17 9 5 3).
The permutation is a composition of 8 cycles of lengths 1, 5, 5, 5, 5, 5, 5, and 1.
Namely, we have the following composition of cycles:
(1)(2 17 9 5 3)(4 18 25 13 7)(6 19 10 21 11)(8 20 26 29 15)(12 22 27 14 23)(16 24 28 30 31)(32).
Omitting the cycles of length one, we have the following composition of cycles:
(2 17 9 5 3)(4 18 25 13 7)(6 19 10 21 11)(8 20 26 29 15)(12 22 27 14 23)(16 24 28 30 31).
So, the order of the permutation is 5.
(17) The case n = 17. The first cycle is a cycle of length 10. This cycle looks like this:
(2 18 26 30 32 33 17 9 5 3).
The permutation is a composition of 6 cycles of lengths 1, 10, 10, 10, 2, and 1.
Namely, we have the following composition of cycles:
(1)(2 18 26 30 32 33 17 9 5 3)(4 19 10 22 28 31 16 25 13 7)(6 20 27 14 24 29 15 8 21 11)(12 23)(34).
Omitting the cycles of length one, we have the following composition of cycles:
(2 18 26 30 32 33 17 9 5 3)(4 19 10 22 28 31 16 25 13 7)(6 20 27 14 24 29 15 8 21 11)(12 23).
So, the order of the permutation is 10.
(18) The case n = 18. The first cycle is a cycle of length 12. This cycle looks like this:
(2 19 10 23 12 24 30 33 17 9 5 3).
The permutation is a composition of 7 cycles of lengths 1, 12, 12, 3, 4, 3, and 1.
Namely, we have the following composition of cycles:
(1)(2 19 10 23 12 24 30 33 17 9 5 3)(4 20 28 32 34 35 18 27 14 25 13 7)(6 21 11)(8 22 29 15)(16 26 31)(36).
Omitting the cycles of length one, we have the following composition of cycles:
(2 19 10 23 12 24 30 33 17 9 5 3)(4 20 28 32 34 35 18 27 14 25 13 7)(6 21 11)(8 22 29 15)(16 26 31).
So, the order of the permutation is 12.
(19) The case n = 19. The first cycle is a cycle of length 36. This cycle looks like this:
(2 20 29 15 8 23 12 25 13 7 4 21 11 6 22 30 34 36 37 19 10 24 31 16 27 14 26 32 35 18 28 33 17 9 5 3).
The permutation is a composition of 3 cycles of lengths 1, 36, and 1. Namely, we
have the following composition of cycles:
(1)(2 20 29 15 8 23 12 25 13 7 4 21 11 6 22 30 34 36 37 19 10 24 31 16 27 14 26 32 35 18 28 33 17 9 5 3)(38).
Omitting the cycles of length one, we have the following composition of cycles:
(2 20 29 15 8 23 12 25 13 7 4 21 11 6 22 30 34 36 37 19 10 24 31 16 27 14 26 32 35 18 28 33 17 9 5 3).
So, the order of the permutation is 36.
We can continue in the same fashion for a very long time (keeping in mind that most of
this text is written by a computer program). But it would be much better to understand
the rule that stands behind this. Unfortunately, I did not find this rule (this does not
mean the rule does not exists and this does not mean that it is hard to find the rule).
Nevertheless, I investigated a lot of cases (using a simple program in Java). Let me
summarize my investigations in a table whose first column contains n for all n from 4 till
1000, the second column contains 2n (the number of interlacing cards), the third column
contains the cycle shape of the permutation like in [1, Example 20.4] and [1, Example 20.5]
(but I put the cycle shape in the ascending order contrary to the book, since it looks more
natural to me), and final forth column contains the order of our permutation (what we
are looking for!).
8
n
2n
cycle-shape
order
4
8
(12 , 32 )
3
5
10
(12 , 2, 6)
6
6
12
(12 , 10)
10
7
14
(12 , 12)
12
8
16
(12 , 2, 43 )
4
9
18
(12 , 82 )
8
10
20
(12 , 18)
18
11
22
(12 , 2, 32 , 62 )
6
12
24
(12 , 112 )
11
13
26
(12 , 4, 20)
20
14
28
(12 , 2, 6, 18)
18
15
30
(12 , 28)
28
16
32
(12 , 56 )
5
17
34
(12 , 2, 103 )
10
18
36
(12 , 32 , 4, 122 )
12
19
38
(12 , 36)
36
20
40
(12 , 2, 123 )
12
21
42
(12 , 202 )
20
22
44
(12 , 143 )
14
23
46
(12 , 2, 43 , 6, 122 )
12
24
48
(12 , 232 )
23
25
50
(12 , 32 , 212 )
21
26
52
(12 , 2, 86 )
8
27
54
(12 , 52)
52
28
56
(12 , 4, 10, 202 )
20
29
58
(12 , 2, 183 )
18
30
60
(12 , 58)
58
31
62
(12 , 60)
60
32
64
(12 , 2, 32 , 69 )
6
33
66
(12 , 4, 125 )
12
34
68
(12 , 66)
66
9
35
70
(12 , 2, 112 , 222 )
22
36
72
(12 , 352 )
35
37
74
(12 , 98 )
9
38
76
(12 , 2, 43 , 203 )
20
39
78
(12 , 32 , 10, 302 )
30
40
80
(12 , 392 )
39
41
82
(12 , 2, 6, 18, 54)
54
42
84
(12 , 82)
82
43
86
(12 , 4, 810 )
8
44
88
(12 , 2, 283 )
28
45
90
(12 , 118 )
11
46
92
(12 , 32 , 127 )
12
47
94
(12 , 2, 56 , 106 )
10
48
96
(12 , 4, 18, 362 )
36
49
98
(12 , 482 )
48
50
100
(12 , 2, 6, 103 , 302 )
30
51
102
(12 , 100)
100
52
104
(12 , 512 )
51
53
106
(12 , 2, 32 , 43 , 62 , 126 )
12
54
108
(12 , 106)
106
55
110
(12 , 363 )
36
56
112
(12 , 2, 363 )
36
57
114
(12 , 284 )
28
58
116
(12 , 4, 112 , 442 )
44
59
118
(12 , 2, 6, 129 )
12
60
120
(12 , 32 , 82 , 244 )
24
61
122
(12 , 10, 110)
110
62
124
(12 , 2, 206 )
20
63
126
(12 , 4, 20, 100)
100
64
128
(12 , 718 )
7
65
130
(12 , 2, 149 )
14
66
132
(12 , 130)
130
67
134
(12 , 32 , 187 )
18
10
68
136
(12 , 2, 43 , 6, 122 , 18, 362 )
36
69
138
(12 , 682 )
68
70
140
(12 , 138)
138
71
142
(12 , 2, 232 , 462 )
46
72
144
(12 , 10, 12, 602 )
60
73
146
(12 , 4, 285 )
28
74
148
(12 , 2, 32 , 62 , 212 , 422 )
42
75
150
(12 , 148)
148
76
152
(12 , 1510 )
15
77
154
(12 , 2, 6, 86 , 244 )
24
78
156
(12 , 4, 56 , 206 )
20
79
158
(12 , 523 )
52
80
160
(12 , 2, 523 )
52
81
162
(12 , 32 , 112 , 334 )
33
82
164
(12 , 162)
162
83
166
(12 , 2, 43 , 103 , 206 )
20
84
168
(12 , 832 )
83
85
170
(12 , 12, 156)
156
86
172
(12 , 2, 6, 189 )
18
87
174
(12 , 172)
172
88
176
(12 , 32 , 4, 122 , 20, 602 )
60
89
178
(12 , 2, 583 )
58
90
180
(12 , 178)
178
91
182
(12 , 180)
180
92
184
(12 , 2, 603 )
60
93
186
(12 , 4, 365 )
36
94
188
(12 , 82 , 10, 404 )
40
95
190
(12 , 2, 32 , 69 , 187 )
18
96
192
(12 , 952 )
95
97
194
(12 , 962 )
96
98
196
(12 , 2, 43 , 1215 )
12
99
198
(12 , 196)
196
100
200
(12 , 992 )
99
11
101
202
(12 , 2, 663 )
66
102
204
(12 , 32 , 28, 842 )
84
103
206
(12 , 4, 2010 )
20
104
208
(12 , 2, 6, 112 , 222 , 662 )
66
105
210
(12 , 10, 18, 902 )
90
106
212
(12 , 210)
210
107
214
(12 , 2, 352 , 702 )
70
108
216
(12 , 4, 143 , 286 )
28
109
218
(12 , 32 , 56 , 1512 )
15
110
220
(12 , 2, 98 , 188 )
18
111
222
(12 , 82 , 12, 248 )
24
112
224
(12 , 376 )
37
113
226
(12 , 2, 43 , 6, 122 , 203 , 602 )
60
114
228
(12 , 226)
226
115
230
(12 , 763 )
76
116
232
(12 , 2, 32 , 62 , 103 , 306 )
30
117
234
(12 , 298 )
29
118
236
(12 , 4, 232 , 922 )
92
119
238
(12 , 2, 392 , 782 )
78
120
240
(12 , 1192 )
119
121
242
(12 , 2410 )
24
122
244
(12 , 2, 6, 18, 54, 162)
162
123
246
(12 , 32 , 4, 122 , 212 , 842 )
84
124
248
(12 , 12, 18, 366 )
36
125
250
(12 , 2, 823 )
82
126
252
(12 , 505 )
50
127
254
(12 , 10, 112 , 1102 )
110
128
256
(12 , 2, 43 , 830 )
8
129
258
(12 , 1616 )
16
130
260
(12 , 32 , 367 )
36
131
262
(12 , 2, 6, 283 , 842 )
84
132
264
(12 , 1312 )
131
133
266
(12 , 4, 525 )
52
12
134
268
(12 , 2, 118 , 228 )
22
135
270
(12 , 268)
268
136
272
(12 , 1352 )
135
137
274
(12 , 2, 32 , 62 , 1221 )
12
138
276
(12 , 4, 10, 2013 )
20
139
278
(12 , 923 )
92
140
280
(12 , 2, 56 , 6, 106 , 306 )
30
141
282
(12 , 704 )
70
142
284
(12 , 943 )
94
143
286
(12 , 2, 43 , 183 , 366 )
36
144
288
(12 , 32 , 202 , 604 )
60
145
290
(12 , 82 , 1362 )
136
146
292
(12 , 2, 486 )
48
147
294
(12 , 292)
292
148
296
(12 , 4, 58, 1162 )
116
149
298
(12 , 2, 6, 103 , 18, 302 , 902 )
90
150
300
(12 , 112 , 12, 1322 )
132
151
302
(12 , 32 , 143 , 426 )
42
152
304
(12 , 2, 1003 )
100
153
306
(12 , 4, 605 )
60
154
308
(12 , 1023 )
102
155
310
(12 , 2, 512 , 1022 )
102
156
312
(12 , 1552 )
155
157
314
(12 , 1562 )
156
158
316
(12 , 2, 32 , 43 , 69 , 1220 )
12
159
318
(12 , 316)
316
160
320
(12 , 10, 28, 1402 )
140
161
322
(12 , 2, 1063 )
106
162
324
(12 , 82 , 18, 724 )
72
163
326
(12 , 4, 125 , 20, 604 )
60
164
328
(12 , 2, 369 )
36
165
330
(12 , 32 , 232 , 694 )
69
166
332
(12 , 3011 )
30
13
167
334
(12 , 2, 6, 369 )
36
168
336
(12 , 4, 66, 1322 )
132
169
338
(12 , 2116 )
21
170
340
(12 , 2, 2812 )
28
171
342
(12 , 56 , 1031 )
10
172
344
(12 , 32 , 212 , 1472 )
147
173
346
(12 , 2, 43 , 112 , 222 , 446 )
44
174
348
(12 , 346)
346
175
350
(12 , 348)
348
176
352
(12 , 2, 6, 129 , 18, 366 )
36
177
354
(12 , 884 )
88
178
356
(12 , 4, 352 , 1402 )
140
179
358
(12 , 2, 32 , 62 , 86 , 2412 )
24
180
360
(12 , 1792 )
179
181
362
(12 , 18, 342)
342
182
364
(12 , 2, 103 , 1103 )
110
183
366
(12 , 4, 98 , 368 )
36
184
368
(12 , 1832 )
183
185
370
(12 , 2, 6, 206 , 604 )
60
186
372
(12 , 32 , 52, 1562 )
156
187
374
(12 , 372)
372
188
376
(12 , 2, 43 , 203 , 1003 )
100
189
378
(12 , 12, 28, 844 )
84
190
380
(12 , 378)
378
191
382
(12 , 2, 718 , 1418 )
14
192
384
(12 , 1912 )
191
193
386
(12 , 32 , 4, 10, 122 , 202 , 302 , 604 )
60
194
388
(12 , 2, 6, 149 , 426 )
42
195
390
(12 , 388)
388
196
392
(12 , 82 , 112 , 884 )
88
197
394
(12 , 2, 1303 )
130
198
396
(12 , 4, 392 , 1562 )
156
199
398
(12 , 449 )
44
14
200
400
(12 , 2, 32 , 62 , 1821 )
18
201
402
(12 , 2002 )
200
202
404
(12 , 56 , 12, 606 )
60
203
406
(12 , 2, 43 , 6, 122 , 18, 362 , 54, 1082 )
108
204
408
(12 , 10, 36, 1802 )
180
205
410
(12 , 2042 )
204
206
412
(12 , 2, 686 )
68
207
414
(12 , 32 , 58, 1742 )
174
208
416
(12 , 4, 82, 1642 )
164
209
418
(12 , 2, 1383 )
138
210
420
(12 , 418)
418
211
422
(12 , 420)
420
212
424
(12 , 2, 6, 232 , 462 , 1382 )
138
213
426
(12 , 4, 810 , 20, 408 )
40
214
428
(12 , 32 , 607 )
60
215
430
(12 , 2, 103 , 123 , 606 )
60
216
432
(12 , 4310 )
43
217
434
(12 , 726 )
72
218
436
(12 , 2, 43 , 2815 )
28
219
438
(12 , 112 , 18, 1982 )
198
220
440
(12 , 736 )
73
221
442
(12 , 2, 32 , 69 , 212 , 428 )
42
222
444
(12 , 442)
442
223
446
(12 , 4, 118 , 448 )
44
224
448
(12 , 2, 1483 )
148
225
450
(12 , 2242 )
224
226
452
(12 , 10, 2022 )
20
227
454
(12 , 2, 1510 , 3010 )
30
228
456
(12 , 32 , 4, 1237 )
12
229
458
(12 , 766 )
76
230
460
(12 , 2, 6, 86 , 18, 244 , 724 )
72
231
462
(12 , 460)
460
232
464
(12 , 2312 )
231
15
233
466
(12 , 2, 43 , 56 , 106 , 2018 )
20
234
468
(12 , 466)
466
235
470
(12 , 32 , 667 )
66
236
472
(12 , 2, 529 )
52
237
474
(12 , 10, 143 , 706 )
70
238
476
(12 , 4, 18, 20, 362 , 1802 )
180
239
478
(12 , 2, 6, 523 , 1562 )
156
240
480
(12 , 2392 )
239
241
482
(12 , 12, 3613 )
36
242
484
(12 , 2, 32 , 62 , 112 , 222 , 334 , 664 )
66
243
486
(12 , 4, 4810 )
48
244
488
(12 , 2432 )
243
245
490
(12 , 2, 1623 )
162
246
492
(12 , 490)
490
247
494
(12 , 82 , 28, 568 )
56
248
496
(12 , 2, 43 , 6, 103 , 122 , 206 , 302 , 604 )
60
249
498
(12 , 32 , 352 , 1054 )
105
250
500
(12 , 1663 )
166
251
502
(12 , 2, 832 , 1662 )
166
252
504
(12 , 2512 )
251
253
506
(12 , 4, 1005 )
100
254
508
(12 , 2, 123 , 1563 )
156
255
510
(12 , 508)
508
256
512
(12 , 32 , 956 )
9
257
514
(12 , 2, 6, 1828 )
18
258
516
(12 , 4, 512 , 2042 )
204
259
518
(12 , 10, 232 , 2302 )
230
260
520
(12 , 2, 1723 )
172
261
522
(12 , 2602 )
260
262
524
(12 , 522)
522
263
526
(12 , 2, 32 , 43 , 62 , 126 , 203 , 606 )
60
264
528
(12 , 56 , 82 , 4012 )
40
265
530
(12 , 112 , 2532 )
253
16
266
532
(12 , 2, 6, 583 , 1742 )
174
267
534
(12 , 12, 202 , 608 )
60
268
536
(12 , 4, 106, 2122 )
212
269
538
(12 , 2, 1783 )
178
270
540
(12 , 32 , 10, 212 , 302 , 2102 )
210
271
542
(12 , 540)
540
272
544
(12 , 2, 1803 )
180
273
546
(12 , 4, 3615 )
36
274
548
(12 , 546)
546
275
550
(12 , 2, 6, 609 )
60
276
552
(12 , 18, 28, 2522 )
252
277
554
(12 , 32 , 3914 )
39
278
556
(12 , 2, 43 , 3615 )
36
279
558
(12 , 556)
556
280
560
(12 , 12, 143 , 846 )
84
281
562
(12 , 2, 86 , 103 , 4012 )
40
282
564
(12 , 562)
562
283
566
(12 , 4, 2820 )
28
284
568
(12 , 2, 32 , 69 , 187 , 547 )
54
285
570
(12 , 2842 )
284
286
572
(12 , 1145 )
114
287
574
(12 , 2, 952 , 1902 )
190
288
576
(12 , 4, 112 , 20, 442 , 2202 )
220
289
578
(12 , 1444 )
144
290
580
(12 , 2, 966 )
96
291
582
(12 , 32 , 82, 2462 )
246
292
584
(12 , 10, 52, 2602 )
260
293
586
(12 , 2, 43 , 6, 1247 )
12
294
588
(12 , 586)
586
295
590
(12 , 56 , 18, 906 )
90
296
592
(12 , 2, 1963 )
196
297
594
(12 , 1484 )
148
298
596
(12 , 32 , 4, 810 , 122 , 2420 )
24
17
299
598
(12 , 2, 992 , 1982 )
198
300
600
(12 , 2992 )
299
301
602
(12 , 2524 )
25
302
604
(12 , 2, 6, 669 )
66
303
606
(12 , 4, 10, 202 , 110, 2202 )
220
304
608
(12 , 3032 )
303
305
610
(12 , 2, 32 , 62 , 283 , 846 )
84
306
612
(12 , 12, 232 , 2762 )
276
307
614
(12 , 612)
612
308
616
(12 , 2, 43 , 2030 )
20
309
618
(12 , 1544 )
154
310
620
(12 , 618)
618
311
622
(12 , 2, 6, 112 , 18, 222 , 662 , 1982 )
198
312
624
(12 , 32 , 118 , 3316 )
33
313
626
(12 , 4, 20, 100, 500)
500
314
628
(12 , 2, 103 , 183 , 906 )
90
315
630
(12 , 82 , 36, 728 )
72
316
632
(12 , 4514 )
45
317
634
(12 , 2, 2103 )
210
318
636
(12 , 4, 718 , 2818 )
28
319
638
(12 , 32 , 127 , 212 , 846 )
84
320
640
(12 , 2, 6, 352 , 702 , 2102 )
210
321
642
(12 , 6410 )
64
322
644
(12 , 2143 )
214
323
646
(12 , 2, 43 , 149 , 2818 )
28
324
648
(12 , 3232 )
323
325
650
(12 , 10, 58, 2902 )
290
326
652
(12 , 2, 32 , 56 , 62 , 106 , 1512 , 3012 )
30
327
654
(12 , 652)
652
328
656
(12 , 4, 130, 2602 )
260
329
658
(12 , 2, 6, 98 , 1832 )
18
330
660
(12 , 658)
658
331
662
(12 , 660)
660
18
332
664
(12 , 2, 86 , 123 , 2424 )
24
333
666
(12 , 32 , 4, 122 , 187 , 3614 )
36
334
668
(12 , 112 , 28, 3082 )
308
335
670
(12 , 2, 376 , 746 )
74
336
672
(12 , 10, 6011 )
60
337
674
(12 , 4814 )
48
338
676
(12 , 2, 43 , 6, 122 , 18, 203 , 362 , 602 , 1802 )
180
339
678
(12 , 676)
676
340
680
(12 , 32 , 4814 )
48
341
682
(12 , 2, 2263 )
226
342
684
(12 , 2231 )
22
343
686
(12 , 4, 6810 )
68
344
688
(12 , 2, 769 )
76
345
690
(12 , 12, 52, 1564 )
156
346
692
(12 , 2303 )
230
347
694
(12 , 2, 32 , 69 , 103 , 3020 )
30
348
696
(12 , 4, 138, 2762 )
276
349
698
(12 , 82 , 202 , 4016 )
40
350
700
(12 , 2, 298 , 588 )
58
351
702
(12 , 700)
700
352
704
(12 , 18, 3619 )
36
353
706
(12 , 2, 43 , 232 , 462 , 926 )
92
354
708
(12 , 32 , 100, 3002 )
300
355
710
(12 , 708)
708
356
712
(12 , 2, 6, 392 , 788 )
78
357
714
(12 , 56 , 112 , 5512 )
55
358
716
(12 , 4, 10, 125 , 202 , 6010 )
60
359
718
(12 , 2, 1192 , 2382 )
238
360
720
(12 , 3592 )
359
361
722
(12 , 32 , 5114 )
51
362
724
(12 , 2, 2430 )
24
363
726
(12 , 4, 20, 285 , 1404 )
140
364
728
(12 , 1216 )
121
19
365
730
(12 , 2, 6, 18, 54, 162, 486)
486
366
732
(12 , 82 , 143 , 5612 )
56
367
734
(12 , 2443 )
244
368
736
(12 , 2, 32 , 43 , 62 , 126 , 212 , 422 , 846 )
84
369
738
(12 , 10, 66, 3302 )
330
370
740
(12 , 2463 )
246
371
742
(12 , 2, 123 , 183 , 3618 )
36
372
744
(12 , 3712 )
371
373
746
(12 , 4, 1485 )
148
374
748
(12 , 2, 6, 823 , 2462 )
246
375
750
(12 , 32 , 106, 3182 )
318
376
752
(12 , 3752 )
375
377
754
(12 , 2, 5015 )
50
378
756
(12 , 4, 1510 , 6010 )
60
379
758
(12 , 756)
756
380
760
(12 , 2, 103 , 112 , 222 , 1106 )
110
381
762
(12 , 3802 )
380
382
764
(12 , 32 , 3621 )
36
383
766
(12 , 2, 43 , 6, 830 , 122 , 2420 )
24
384
768
(12 , 12, 58, 3482 )
348
385
770
(12 , 3842 )
384
386
772
(12 , 2, 1648 )
16
387
774
(12 , 772)
772
388
776
(12 , 4, 56 , 2037 )
20
389
778
(12 , 2, 32 , 62 , 3621 )
36
390
780
(12 , 18, 202 , 1804 )
180
391
782
(12 , 10, 352 , 7010 )
70
392
784
(12 , 2, 6, 18, 283 , 842 , 2522 )
252
393
786
(12 , 4, 5215 )
52
394
788
(12 , 786)
786
395
790
(12 , 2, 1312 , 2622 )
262
396
792
(12 , 32 , 284 , 848 )
84
397
794
(12 , 12, 6013 )
60
20
398
796
(12 , 2, 43 , 5215 )
52
399
798
(12 , 796)
796
400
800
(12 , 82 , 232 , 1844 )
184
401
802
(12 , 2, 6, 118 , 228 , 668 )
66
402
804
(12 , 98 , 10, 908 )
90
403
806
(12 , 32 , 4, 112 , 122 , 334 , 442 , 1324 )
132
404
808
(12 , 2, 2683 )
268
405
810
(12 , 4042 )
404
406
812
(12 , 2703 )
270
407
814
(12 , 2, 1352 , 2702 )
270
408
816
(12 , 4, 162, 3242 )
324
409
818
(12 , 143 , 18, 1266 )
126
410
820
(12 , 2, 32 , 69 , 1263 )
12
411
822
(12 , 820)
820
412
824
(12 , 4112 )
411
413
826
(12 , 2, 43 , 103 , 2039 )
20
414
828
(12 , 826)
826
415
830
(12 , 828)
828
416
832
(12 , 2, 929 )
92
417
834
(12 , 32 , 82 , 212 , 244 , 1684 )
168
418
836
(12 , 4, 832 , 3322 )
332
419
838
(12 , 2, 56 , 6, 106 , 18, 306 , 906 )
90
420
840
(12 , 4192 )
419
421
842
(12 , 28, 812)
812
422
844
(12 , 2, 7012 )
70
423
846
(12 , 4, 125 , 1565 )
156
424
848
(12 , 32 , 10, 302 , 110, 3302 )
330
425
850
(12 , 2, 949 )
94
426
852
(12 , 112 , 36, 3962 )
396
427
854
(12 , 852)
852
428
856
(12 , 2, 43 , 6, 122 , 189 , 3618 )
36
429
858
(12 , 4282 )
428
430
860
(12 , 858)
858
21
431
862
(12 , 2, 32 , 62 , 206 , 6012 )
60
432
864
(12 , 4312 )
431
433
866
(12 , 4, 1725 )
172
434
868
(12 , 2, 86 , 1366 )
136
435
870
(12 , 10, 392 , 3902 )
390
436
872
(12 , 12, 66, 1326 )
132
437
874
(12 , 2, 6, 4818 )
48
438
876
(12 , 32 , 4, 122 , 20, 602 , 100, 3002 )
300
439
878
(12 , 876)
876
440
880
(12 , 2, 2923 )
292
441
882
(12 , 5516 )
55
442
884
(12 , 882)
882
443
886
(12 , 2, 43 , 583 , 1166 )
116
444
888
(12 , 4432 )
443
445
890
(12 , 32 , 718 , 2136 )
21
446
892
(12 , 2, 6, 103 , 18, 302 , 54, 902 , 2702 )
270
447
894
(12 , 18, 232 , 4142 )
414
448
896
(12 , 4, 178, 3562 )
356
449
898
(12 , 2, 112 , 123 , 222 , 1326 )
132
450
900
(12 , 56 , 28, 1406 )
140
451
902
(12 , 82 , 52, 1048 )
104
452
904
(12 , 2, 32 , 62 , 149 , 4218 )
42
453
906
(12 , 4, 1805 )
180
454
908
(12 , 906)
906
455
910
(12 , 2, 6, 1003 , 3002 )
300
456
912
(12 , 9110 )
91
457
914
(12 , 10, 82, 4102 )
410
458
916
(12 , 2, 43 , 6015 )
60
459
918
(12 , 32 , 130, 3902 )
390
460
920
(12 , 1536 )
153
461
922
(12 , 2, 1029 )
102
462
924
(12 , 12, 352 , 4202 )
420
463
926
(12 , 4, 20, 365 , 1804 )
180
22
464
928
(12 , 2, 6, 512 , 1028 )
102
465
930
(12 , 4642 )
464
466
932
(12 , 32 , 187 , 212 , 1266 )
126
467
934
(12 , 2, 1552 , 3102 )
310
468
936
(12 , 4, 810 , 10, 202 , 4020 )
40
469
938
(12 , 1178 )
117
470
940
(12 , 2, 1566 )
156
471
942
(12 , 940)
940
472
944
(12 , 112 , 202 , 2204 )
220
473
946
(12 , 2, 32 , 43 , 69 , 1220 , 187 , 3614 )
36
474
948
(12 , 946)
946
475
950
(12 , 98 , 12, 3624 )
36
476
952
(12 , 2, 3163 )
316
477
954
(12 , 6814 )
68
478
956
(12 , 4, 952 , 3802 )
380
479
958
(12 , 2, 103 , 283 , 1406 )
140
480
960
(12 , 32 , 682 , 2044 )
204
481
962
(12 , 56 , 1556 )
155
482
964
(12 , 2, 6, 1063 , 3182 )
318
483
966
(12 , 4, 9610 )
96
484
968
(12 , 4832 )
483
485
970
(12 , 2, 86 , 183 , 7212 )
72
486
972
(12 , 1945 )
194
487
974
(12 , 32 , 1387 )
138
488
976
(12 , 2, 43 , 1215 , 203 , 6012 )
60
489
978
(12 , 4882 )
488
490
980
(12 , 10, 118 , 1108 )
110
491
982
(12 , 2, 6, 3627 )
36
492
984
(12 , 4912 )
491
493
986
(12 , 4, 1965 )
196
494
988
(12 , 2, 32 , 62 , 232 , 462 , 694 , 1384 )
138
495
990
(12 , 112 , 143 , 1546 )
154
496
992
(12 , 4952 )
495
23
497
994
(12 , 2, 3033 )
30
498
996
(12 , 4, 992 , 3962 )
396
499
998
(12 , 3323 )
332
500
1000 (12 , 2, 6, 18, 3627 )
36
501
1002 (12 , 32 , 10, 127 , 302 , 6014 )
60
502
1004 (12 , 82 , 58, 2324 )
232
503
1006 (12 , 2, 43 , 663 , 1326 )
132
504
1008 (12 , 18, 52, 4682 )
468
505
1010 (12 , 5042 )
504
506
1012 (12 , 2, 2116 , 4216 )
42
507
1014 (12 , 9211 )
92
508
1016 (12 , 32 , 4, 122 , 285 , 8410 )
84
509
1018 (12 , 2, 6, 2812 , 848 )
84
510
1020 (12 , 1018)
1018
511
1022 (12 , 3403 )
340
512
1024 (12 , 2, 56 , 1099 )
10
513
1026 (12 , 4, 2051 )
20
514
1028 (12 , 12, 392 , 1566 )
156
515
1030 (12 , 2, 32 , 62 , 212 , 422 , 1472 , 2942 )
294
516
1032 (12 , 5152 )
515
517
1034 (12 , 2584 )
258
518
1036 (12 , 2, 43 , 6, 112 , 122 , 222 , 446 , 662 , 1324 )
132
519
1038 (12 , 82 , 60, 1208 )
120
520
1040
(12 , 5192 )
519
521
1042 (12 , 2, 3463 )
346
522
1044 (12 , 32 , 148, 4442 )
444
523
1046 (12 , 4, 10, 18, 202 , 362 , 902 , 1804 )
180
524
1048 (12 , 2, 3483 )
348
525
1050 (12 , 2624 )
262
526
1052 (12 , 3503 )
350
527
1054 (12 , 2, 6, 129 , 18, 366 , 54, 1086 )
108
528
1056 (12 , 4, 210, 4202 )
420
529
1058 (12 , 32 , 1570 )
15
24
530
1060 (12 , 2, 8812 )
88
531
1062 (12 , 1060)
1060
532
1064 (12 , 5312 )
531
533
1066 (12 , 2, 43 , 352 , 702 , 1406 )
140
534
1068 (12 , 10, 482 , 2404 )
240
535
1070 (12 , 3563 )
356
536
1072 (12 , 2, 32 , 69 , 86 , 2440 )
24
537
1074 (12 , 28, 36, 2524 )
252
538
1076 (12 , 4, 143 , 20, 286 , 1406 )
140
539
1078 (12 , 2, 1792 , 3582 )
358
540
1080 (12 , 12, 82, 4922 )
492
541
1082 (12 , 112 , 232 , 2534 )
253
542
1084 (12 , 2, 183 , 3423 )
342
543
1086 (12 , 32 , 4, 56 , 122 , 1512 , 206 , 6012 )
60
544
1088 (12 , 5432 )
543
545
1090 (12 , 2, 6, 103 , 302 , 1103 , 3302 )
330
546
1092 (12 , 1090)
1090
547
1094 (12 , 3643 )
364
548
1096 (12 , 2, 43 , 98 , 188 , 3624 )
36
549
1098 (12 , 2744 )
274
550
1100 (12 , 32 , 523 , 1566 )
156
551
1102 (12 , 2, 1832 , 3662 )
366
552
1104 (12 , 2938 )
29
553
1106
(12 , 4, 810 , 125 , 2440 )
24
554
1108 (12 , 2, 6, 18, 206 , 604 , 1804 )
180
555
1110 (12 , 1108)
1108
556
1112 (12 , 10, 10011 )
100
557
1114 (12 , 2, 32 , 62 , 523 , 1566 )
156
558
1116 (12 , 4, 376 , 1486 )
148
559
1118 (12 , 1116)
1116
560
1120 (12 , 2, 3723 )
372
561
1122 (12 , 18, 58, 5222 )
522
562
1124 (12 , 1122)
1122
25
563
1126 (12 , 2, 43 , 6, 122 , 203 , 602 , 1003 , 3002 )
300
564
1128 (12 , 32 , 112 , 212 , 334 , 2314 )
231
565
1130 (12 , 5642 )
564
566
1132 (12 , 2, 123 , 283 , 8412 )
84
567
1134 (12 , 10, 512 , 5102 )
510
568
1136 (12 , 4, 226, 4522 )
452
569
1138 (12 , 2, 3783 )
378
570
1140 (12 , 82 , 66, 2644 )
264
571
1142 (12 , 32 , 1627 )
162
572
1144 (12 , 2, 6, 718 , 1418 , 4218 )
42
573
1146 (12 , 4, 7615 )
76
574
1148 (12 , 56 , 36, 1806 )
180
575
1150 (12 , 2, 1912 , 3822 )
382
576
1152 (12 , 5752 )
575
577
1154 (12 , 2884 )
288
578
1156 (12 , 2, 32 , 43 , 62 , 103 , 126 , 206 , 306 , 6012 )
60
579
1158 (12 , 118 , 12, 1328 )
132
580
1160 (12 , 18, 60, 1806 )
180
581
1162 (12 , 2, 6, 149 , 18, 426 , 1266 )
126
582
1164 (12 , 1667 )
166
583
1166 (12 , 4, 298 , 1168 )
116
584
1168 (12 , 2, 3883 )
388
585
1170 (12 , 32 , 832 , 2494 )
249
586
1172
(12 , 1170)
1170
587
1174 (12 , 2, 86 , 112 , 222 , 8812 )
88
588
1176 (12 , 4, 20, 232 , 922 , 4602 )
460
589
1178 (12 , 10, 106, 5302 )
530
590
1180 (12 , 2, 6, 1303 , 3902 )
390
591
1182 (12 , 2365 )
236
592
1184 (12 , 32 , 127 , 1567 )
156
593
1186 (12 , 2, 43 , 392 , 782 , 1566 )
156
594
1188 (12 , 1186)
1186
595
1190 (12 , 202 , 28, 1408 )
140
26
596
1192 (12 , 2, 4427 )
44
597
1194 (12 , 2984 )
298
598
1196 (12 , 4, 1192 , 4762 )
476
599
1198 (12 , 2, 32 , 69 , 1863 )
18
600
1200 (12 , 10, 363 , 1806 )
180
601
1202 (12 , 3004 )
300
602
1204 (12 , 2, 2006 )
200
603
1206 (12 , 4, 2450 )
24
604
1208 (12 , 82 , 352 , 2804 )
280
605
1210 (12 , 2, 56 , 106 , 123 , 6018 )
60
606
1212 (12 , 32 , 172, 5162 )
516
607
1214 (12 , 1212)
1212
608
1216 (12 , 2, 43 , 6, 122 , 18, 362 , 54, 1082 , 162, 3242 )
324
609
1218 (12 , 1528 )
152
610
1220 (12 , 112 , 52, 5722 )
572
611
1222 (12 , 2, 103 , 363 , 1806 )
180
612
1224 (12 , 6112 )
611
613
1226 (12 , 32 , 4, 122 , 20, 212 , 602 , 842 , 4202 )
420
614
1228 (12 , 2, 2046 )
204
615
1230 (12 , 1228)
1228
616
1232 (12 , 6152 )
615
617
1234 (12 , 2, 6, 686 , 2044 )
204
618
1236 (12 , 4, 125 , 18, 3632 )
36
(12 , 1236)
619
1238
1236
620
1240 (12 , 2, 32 , 62 , 583 , 1746 )
174
621
1242 (12 , 82 , 98 , 7216 )
72
622
1244 (12 , 10, 284 , 1408 )
140
623
1246 (12 , 2, 43 , 823 , 1646 )
164
624
1248 (12 , 143 , 2843 )
28
625
1250 (12 , 1568 )
156
626
1252 (12 , 2, 6, 1389 )
138
627
1254 (12 , 32 , 178, 5342 )
534
628
1256 (12 , 4, 505 , 10010 )
100
27
629
1258 (12 , 2, 4183 )
418
630
1260 (12 , 1258)
1258
631
1262 (12 , 12, 4826 )
48
632
1264 (12 , 2, 4203 )
420
633
1266 (12 , 4, 10, 112 , 202 , 442 , 1102 , 2204 )
220
634
1268 (12 , 32 , 1807 )
180
635
1270 (12 , 2, 6, 18, 232 , 462 , 1382 , 4142 )
414
636
1272 (12 , 56 , 2062 )
20
637
1274 (12 , 18, 66, 1986 )
198
638
1276 (12 , 2, 43 , 830 , 203 , 4024 )
40
639
1278 (12 , 1276)
1276
640
1280 (12 , 6392 )
639
641
1282 (12 , 2, 32 , 62 , 6021 )
60
642
1284 (12 , 1282)
1282
643
1286 (12 , 4, 1680 )
16
644
1288 (12 , 2, 6, 103 , 129 , 302 , 6018 )
60
645
1290 (12 , 1618 )
161
646
1292 (12 , 1290)
1290
647
1294 (12 , 2, 4310 , 8610 )
86
648
1296 (12 , 32 , 4, 122 , 3635 )
36
649
1298 (12 , 6482 )
648
650
1300 (12 , 2, 7218 )
72
651
1302 (12 , 1300)
1300
652
1304
(12 , 6512 )
653
1306 (12 , 2, 43 , 6, 122 , 2815 , 8410 )
84
654
1308 (12 , 1306)
1306
655
1310 (12 , 32 , 82 , 10, 244 , 302 , 404 , 1208 )
120
656
1312 (12 , 2, 112 , 183 , 222 , 1986 )
198
657
1314 (12 , 12, 100, 3004 )
300
658
1316 (12 , 4, 1312 , 5242 )
524
659
1318 (12 , 2, 736 , 1466 )
146
660
1320 (12 , 6592 )
659
661
1322 (12 , 6022 )
60
651
28
662
1324 (12 , 2, 32 , 69 , 187 , 212 , 428 , 1266 )
126
663
1326 (12 , 4, 20, 525 , 2604 )
260
664
1328 (12 , 2216 )
221
665
1330 (12 , 2, 4423 )
442
666
1332 (12 , 10, 110, 1210)
1210
667
1334 (12 , 56 , 143 , 7018 )
70
668
1336 (12 , 2, 43 , 118 , 228 , 4424 )
44
669
1338 (12 , 32 , 952 , 2854 )
285
670
1340 (12 , 12, 512 , 2046 )
204
671
1342 (12 , 2, 6, 1483 , 4442 )
444
672
1344 (12 , 82 , 392 , 3124 )
312
673
1346 (12 , 4, 2685 )
268
674
1348 (12 , 2, 2246 )
224
675
1350 (12 , 18, 352 , 6302 )
630
676
1352 (12 , 32 , 9614 )
96
677
1354 (12 , 2, 103 , 2066 )
20
678
1356 (12 , 4, 1352 , 5402 )
540
679
1358 (12 , 112 , 58, 6382 )
638
680
1360 (12 , 2, 6, 1510 , 3040 )
30
681
1362 (12 , 6802 )
680
682
1364 (12 , 232 , 28, 6442 )
644
683
1366 (12 , 2, 32 , 43 , 62 , 12111 )
12
684
1368 (12 , 6832 )
683
(12 , 36, 1332)
685
1370
1332
686
1372 (12 , 2, 7618 )
76
687
1374 (12 , 1372)
1372
688
1376 (12 , 4, 10, 2013 , 10011 )
100
689
1378 (12 , 2, 6, 86 , 18, 244 , 54, 724 , 2164 )
216
690
1380 (12 , 32 , 196, 5882 )
588
691
1382 (12 , 1380)
1380
692
1384 (12 , 2, 4603 )
460
693
1386 (12 , 4, 9215 )
92
694
1388 (12 , 98 , 1873 )
18
29
695
1390 (12 , 2, 2312 , 4622 )
462
696
1392 (12 , 12, 106, 6362 )
636
697
1394 (12 , 32 , 9914 )
99
698
1396 (12 , 2, 43 , 56 , 6, 106 , 122 , 2018 , 306 , 6012 )
60
699
1398 (12 , 718 , 10, 7018 )
70
700
1400 (12 , 2336 )
233
701
1402 (12 , 2, 4663 )
466
702
1404 (12 , 112 , 60, 6602 )
660
703
1406 (12 , 4, 704 , 1408 )
140
704
1408 (12 , 2, 32 , 62 , 6621 )
66
705
1410 (12 , 7042 )
704
706
1412 (12 , 82 , 82, 3284 )
328
707
1414 (12 , 2, 6, 529 , 1566 )
156
708
1416 (12 , 4, 943 , 1886 )
188
709
1418 (12 , 12, 3639 )
36
710
1420 (12 , 2, 103 , 149 , 7018 )
70
711
1422 (12 , 32 , 212 , 28, 8416 )
84
712
1424 (12 , 2376 )
237
713
1426 (12 , 2, 43 , 183 , 203 , 366 , 1806 )
180
714
1428 (12 , 1426)
1426
715
1430 (12 , 8417 )
84
716
1432 (12 , 2, 6, 18, 523 , 1562 , 4682 )
468
717
1434 (12 , 1798 )
179
(12 , 32 , 4, 122 , 2010 , 6020 )
718
1436
719
1438 (12 , 2, 2392 , 4782 )
478
720
1440 (12 , 7192 )
719
721
1442 (12 , 10, 13011 )
130
722
1444 (12 , 2, 123 , 3639 )
36
723
1446 (12 , 4, 810 , 13610 )
136
724
1448 (12 , 7232 )
723
725
1450 (12 , 2, 32 , 69 , 112 , 222 , 334 , 6618 )
66
726
1452 (12 , 1450)
1450
727
1454 (12 , 1452)
1452
30
60
728
1456 (12 , 2, 43 , 4830 )
48
729
1458 (12 , 56 , 232 , 11512 )
115
730
1460 (12 , 4863 )
486
731
1462 (12 , 2, 2432 , 4862 )
486
732
1464 (12 , 32 , 10, 187 , 302 , 9014 )
90
733
1466 (12 , 4, 2925 )
292
734
1468 (12 , 2, 6, 1629 )
162
735
1470 (12 , 12, 284 , 8416 )
84
736
1472 (12 , 2456 )
245
737
1474 (12 , 2, 4903 )
490
738
1476 (12 , 4, 20, 58, 1162 , 5802 )
580
739
1478 (12 , 32 , 2107 )
210
740
1480 (12 , 2, 86 , 283 , 5624 )
56
741
1482 (12 , 3704 )
370
742
1484 (12 , 1482)
1482
743
1486 (12 , 2, 43 , 6, 103 , 122 , 18, 206 , 302 , 362 , 604 , 902 , 1804 ) 180
744
1488 (12 , 7432 )
743
745
1490 (12 , 7442 )
744
746
1492 (12 , 2, 32 , 62 , 352 , 702 , 1054 , 2104 )
210
747
1494 (12 , 1492)
1492
748
1496 (12 , 4, 112 , 125 , 442 , 13210 )
132
749
1498 (12 , 2, 1669 )
166
750
1500 (12 , 1498)
1498
751
1502
(12 , 18, 392 , 2346 )
234
752
1504 (12 , 2, 6, 832 , 1662 , 4982 )
498
753
1506 (12 , 32 , 4, 122 , 143 , 286 , 426 , 8412 )
84
754
1508 (12 , 10, 682 , 3404 )
340
755
1510 (12 , 2, 2512 , 5022 )
502
756
1512 (12 , 7552 )
755
757
1514 (12 , 82 , 118 , 8816 )
88
758
1516 (12 , 2, 43 , 10015 )
100
759
1518 (12 , 202 , 36, 1808 )
180
760
1520 (12 , 32 , 56 , 1512 , 212 , 10512 )
105
31
761
1522 (12 , 2, 6, 129 , 1569 )
156
762
1524 (12 , 1522)
1522
763
1526 (12 , 4, 20, 6025 )
60
764
1528 (12 , 2, 5083 )
508
765
1530 (12 , 10, 138, 6902 )
690
766
1532 (12 , 1530)
1530
767
1534 (12 , 2, 32 , 62 , 956 , 1856 )
18
768
1536 (12 , 4, 1023 , 2046 )
204
769
1538 (12 , 28, 52, 3644 )
364
770
1540 (12 , 2, 6, 1828 , 5419 )
54
771
1542 (12 , 112 , 6623 )
66
772
1544 (12 , 7712 )
771
773
1546 (12 , 2, 43 , 512 , 1022 , 2046 )
204
774
1548 (12 , 32 , 82 , 127 , 2460 )
24
775
1550 (12 , 1548)
1548
776
1552 (12 , 2, 103 , 232 , 462 , 2306 )
230
777
1554 (12 , 1948 )
194
778
1556 (12 , 4, 1552 , 6202 )
620
779
1558 (12 , 2, 6, 1723 , 5162 )
516
780
1560 (12 , 7792 )
779
781
1562 (12 , 32 , 376 , 11112 )
111
782
1564 (12 , 2, 2606 )
260
783
1566 (12 , 4, 15610 )
156
(12 , 7832 )
784
1568
783
785
1570 (12 , 2, 5223 )
522
786
1572 (12 , 1570)
1570
787
1574 (12 , 10, 12, 602 , 110, 6602 )
660
788
1576 (12 , 2, 32 , 43 , 69 , 1220 , 203 , 6020 )
60
789
1578 (12 , 18, 82, 7382 )
738
790
1580 (12 , 5263 )
526
791
1582 (12 , 2, 56 , 86 , 106 , 4036 )
40
792
1584 (12 , 7912 )
791
793
1586 (12 , 4, 3165 )
316
32
794
1588 (12 , 2, 112 , 222 , 2532 , 5062 )
506
795
1590 (12 , 32 , 226, 6782 )
678
796
1592 (12 , 143 , 36, 2526 )
252
797
1594 (12 , 2, 6, 18, 583 , 1742 , 5222 )
522
798
1596 (12 , 4, 10, 202 , 285 , 14010 )
140
799
1598 (12 , 5323 )
532
800
1600 (12 , 2, 123 , 206 , 6024 )
60
801
1602 (12 , 4004 )
400
802
1604 (12 , 32 , 763 , 2286 )
228
803
1606 (12 , 2, 43 , 1063 , 2126 )
212
804
1608 (12 , 8032 )
803
805
1610 (12 , 2018 )
201
806
1612 (12 , 2, 6, 1783 , 5342 )
534
807
1614 (12 , 5231 )
52
808
1616 (12 , 4, 810 , 18, 362 , 7220 )
72
809
1618 (12 , 2, 32 , 62 , 103 , 212 , 306 , 422 , 2106 )
210
810
1620 (12 , 1618)
1618
811
1622 (12 , 1620)
1620
812
1624 (12 , 2, 5403 )
540
813
1626 (12 , 4, 125 , 20, 604 , 100, 3004 )
300
814
1628 (12 , 5423 )
542
815
1630 (12 , 2, 6, 1809 )
180
816
1632 (12 , 32 , 298 , 8716 )
87
(12 , 112 , 352 , 3854 )
817
1634
385
818
1636 (12 , 2, 43 , 3645 )
36
819
1638 (12 , 1636)
1636
820
1640 (12 , 10, 148, 7402 )
740
821
1642 (12 , 2, 5463 )
546
822
1644 (12 , 56 , 52, 2606 )
260
823
1646 (12 , 32 , 4, 122 , 232 , 694 , 922 , 2764 )
276
824
1648 (12 , 2, 6, 18, 609 , 1806 )
180
825
1650 (12 , 82 , 4834 )
48
826
1652 (12 , 718 , 12, 8418 )
84
33
827
1654 (12 , 2, 183 , 283 , 2526 )
252
828
1656 (12 , 4, 3011 , 6022 )
60
829
1658 (12 , 9218 )
92
830
1660 (12 , 2, 32 , 62 , 3914 , 7814 )
78
831
1662 (12 , 10, 1510 , 3050 )
30
832
1664 (12 , 8312 )
831
833
1666 (12 , 2, 43 , 6, 122 , 3645 )
36
834
1668 (12 , 1666)
1666
835
1670 (12 , 1668)
1668
836
1672 (12 , 2, 5563 )
556
837
1674 (12 , 32 , 1192 , 3574 )
357
838
1676 (12 , 4, 20, 66, 1322 , 6602 )
660
839
1678 (12 , 2, 123 , 149 , 8418 )
84
840
1680 (12 , 98 , 112 , 9916 )
99
841
1682 (12 , 202 , 8202 )
820
842
1684 (12 , 2, 6, 86 , 103 , 244 , 302 , 4012 , 1208 )
120
843
1686 (12 , 4, 2116 , 8416 )
84
844
1688 (12 , 32 , 2470 )
24
845
1690 (12 , 2, 5623 )
562
846
1692 (12 , 118 , 18, 1988 )
198
847
1694 (12 , 1692)
1692
848
1696 (12 , 2, 43 , 2860 )
28
849
1698 (12 , 8482 )
848
850
1700
(12 , 5663 )
566
851
1702 (12 , 2, 32 , 69 , 187 , 547 , 1627 )
162
852
1704 (12 , 12, 130, 7802 )
780
853
1706 (12 , 4, 56 , 1031 , 2068 )
20
854
1708 (12 , 2, 2846 )
284
855
1710 (12 , 2447 )
244
856
1712 (12 , 28, 58, 8122 )
812
857
1714 (12 , 2, 11415 )
114
858
1716 (12 , 32 , 4, 122 , 212 , 842 , 1472 , 5882 )
588
859
1718 (12 , 82 , 100, 2008 )
200
34
860
1720 (12 , 2, 6, 952 , 1902 , 5702 )
570
861
1722 (12 , 2158 )
215
862
1724 (12 , 5743 )
574
863
1726 (12 , 2, 43 , 112 , 203 , 222 , 446 , 2206 )
220
864
1728 (12 , 10, 523 , 2606 )
260
865
1730 (12 , 32 , 127 , 187 , 3642 )
36
866
1732 (12 , 2, 14412 )
144
867
1734 (12 , 1732)
1732
868
1736 (12 , 4, 346, 6922 )
692
869
1738 (12 , 2, 6, 9618 )
96
870
1740 (12 , 232 , 36, 8282 )
828
871
1742 (12 , 1740)
1740
872
1744 (12 , 2, 32 , 62 , 823 , 2466 )
246
873
1746 (12 , 4, 3485 )
348
874
1748 (12 , 1746)
1746
875
1750 (12 , 2, 103 , 523 , 2606 )
260
876
1752 (12 , 82 , 512 , 4084 )
408
877
1754 (12 , 14612 )
146
878
1756 (12 , 2, 43 , 6, 1247 , 18, 3632 )
36
879
1758 (12 , 32 , 505 , 15010 )
150
880
1760 (12 , 8792 )
879
881
1762 (12 , 2, 5863 )
586
882
1764 (12 , 143 , 202 , 14012 )
140
883
1766
(12 , 4, 8820 )
88
884
1768 (12 , 2, 56 , 106 , 183 , 9018 )
90
885
1770 (12 , 28, 60, 4204 )
420
886
1772 (12 , 32 , 10, 112 , 302 , 334 , 1102 , 3304 )
330
887
1774 (12 , 2, 6, 1963 , 5882 )
588
888
1776 (12 , 4, 20, 352 , 14012 )
140
889
1778 (12 , 7424 )
74
890
1780 (12 , 2, 14812 )
148
891
1782 (12 , 12, 682 , 2048 )
204
892
1784 (12 , 8912 )
891
35
893
1786 (12 , 2, 32 , 43 , 62 , 830 , 126 , 2460 )
24
894
1788 (12 , 1786)
1786
895
1790 (12 , 5963 )
596
896
1792 (12 , 2, 6, 992 , 1988 )
198
897
1794 (12 , 10, 162, 8102 )
810
898
1796 (12 , 4, 1792 , 7162 )
716
899
1798 (12 , 2, 2992 , 5982 )
598
900
1800 (12 , 32 , 1616 , 4832 )
48
901
1802 (12 , 2572 )
25
902
1804 (12 , 2, 2524 , 5024 )
50
903
1806 (12 , 4, 18, 362 , 342, 6842 )
684
904
1808 (12 , 12, 138, 2766 )
276
905
1810 (12 , 2, 6, 18, 669 , 1986 )
198
906
1812 (12 , 3625 )
362
907
1814 (12 , 32 , 212 , 367 , 2526 )
252
908
1816 (12 , 2, 43 , 103 , 206 , 1103 , 2206 )
220
909
1818 (12 , 112 , 392 , 4294 )
429
910
1820 (12 , 82 , 106, 4244 )
424
911
1822 (12 , 2, 3032 , 6062 )
606
912
1824 (12 , 9112 )
911
913
1826 (12 , 4, 98 , 20, 368 , 1808 )
180
914
1828 (12 , 2, 32 , 69 , 283 , 8420 )
84
915
1830 (12 , 56 , 58, 2906 )
290
916
1832
(12 , 3056 )
305
917
1834 (12 , 2, 123 , 232 , 462 , 2766 )
276
918
1836 (12 , 4, 1832 , 7322 )
732
919
1838 (12 , 10, 832 , 8302 )
830
920
1840 (12 , 2, 6123 )
612
921
1842 (12 , 32 , 1312 , 3934 )
393
922
1844 (12 , 18, 482 , 14412 )
144
923
1846 (12 , 2, 43 , 6, 122 , 2030 , 6020 )
60
924
1848 (12 , 9232 )
923
925
1850 (12 , 143 , 6023 )
602
36
926
1852 (12 , 2, 15412 )
154
927
1854 (12 , 82 , 363 , 7224 )
72
928
1856 (12 , 32 , 4, 122 , 525 , 15610 )
156
929
1858 (12 , 2, 6183 )
618
930
1860 (12 , 10, 12, 602 , 156, 7802 )
780
931
1862 (12 , 1860)
1860
932
1864 (12 , 2, 6, 112 , 18, 222 , 54, 662 , 1982 , 5942 )
594
933
1866 (12 , 4, 3725 )
372
934
1868 (12 , 1866)
1866
935
1870 (12 , 2, 32 , 62 , 118 , 228 , 3316 , 6616 )
66
936
1872 (12 , 9352 )
935
937
1874 (12 , 9362 )
936
938
1876 (12 , 2, 43 , 203 , 1003 , 5003 )
500
939
1878 (12 , 1876)
1876
940
1880 (12 , 9392 )
939
941
1882 (12 , 2, 6, 103 , 189 , 302 , 9018 )
90
942
1884 (12 , 32 , 268, 8042 )
804
943
1886 (12 , 4, 125 , 285 , 8420 )
84
944
1888 (12 , 2, 86 , 363 , 7224 )
72
945
1890 (12 , 4724 )
472
946
1892 (12 , 56 , 6031 )
60
947
1894 (12 , 2, 4514 , 9014 )
90
948
1896 (12 , 4, 378, 7562 )
756
949
1898
(12 , 32 , 13514 )
135
950
1900 (12 , 2, 6, 2109 )
210
951
1902 (12 , 1900)
1900
952
1904 (12 , 10, 172, 8602 )
860
953
1906 (12 , 2, 43 , 718 , 1418 , 2854 )
28
954
1908 (12 , 1906)
1906
955
1910 (12 , 112 , 82, 9022 )
902
956
1912 (12 , 2, 32 , 62 , 1221 , 212 , 422 , 8418 )
84
957
1914 (12 , 2398 )
239
958
1916 (12 , 4, 1912 , 7642 )
764
37
959
1918 (12 , 2, 6, 18, 352 , 702 , 2102 , 6302 )
630
960
1920 (12 , 18, 100, 9002 )
900
961
1922 (12 , 82 , 284 , 5632 )
56
962
1924 (12 , 2, 6430 )
64
963
1926 (12 , 32 , 4, 10, 122 , 2013 , 302 , 6026 )
60
964
1928 (12 , 202 , 232 , 4604 )
460
965
1930 (12 , 2, 2149 )
214
966
1932 (12 , 1930)
1930
967
1934 (12 , 6443 )
644
968
1936 (12 , 2, 43 , 6, 122 , 149 , 2818 , 426 , 8412 )
84
969
1938 (12 , 12, 148, 4444 )
444
970
1940 (12 , 32 , 923 , 2766 )
276
971
1942 (12 , 2, 3232 , 6462 )
646
972
1944 (12 , 28, 66, 9242 )
924
973
1946 (12 , 4, 3885 )
388
974
1948 (12 , 2, 103 , 583 , 2906 )
290
975
1950 (12 , 1948)
1948
976
1952 (12 , 9752 )
975
977
1954 (12 , 2, 32 , 56 , 69 , 106 , 1512 , 3054 )
30
978
1956 (12 , 4, 810 , 112 , 442 , 8820 )
88
979
1958 (12 , 18, 512 , 3066 )
306
980
1960 (12 , 2, 6523 )
652
981
1962 (12 , 36, 52, 4684 )
468
(12 , 12, 1510 , 6030 )
982
1964
60
983
1966 (12 , 2, 43 , 1303 , 2606 )
260
984
1968 (12 , 32 , 704 , 2108 )
210
985
1970 (12 , 10, 178, 8902 )
890
986
1972 (12 , 2, 6, 98 , 18105 )
18
987
1974 (12 , 1972)
1972
988
1976 (12 , 4, 20, 392 , 1562 , 7802 )
780
989
1978 (12 , 2, 6583 )
658
990
1980 (12 , 1978)
1978
991
1982 (12 , 32 , 943 , 2826 )
282
38
992
1984 (12 , 2, 6603 )
660
993
1986 (12 , 4, 4445 )
44
994
1988 (12 , 1986)
1986
995
1990 (12 , 2, 6, 86 , 129 , 2476 )
24
996
1992 (12 , 10, 18011 )
180
997
1994 (12 , 9962 )
996
998
1996 (12 , 2, 32 , 43 , 62 , 126 , 1821 , 3642 )
36
999
1998 (12 , 1996)
1996
1000 2000 (12 , 3336 )
333
References
[1] M. Liebeck, A concise introduction to pure mathematics
Third edition (2010), CRC Press
[2] MATH08004, Group Theory: an Introduction to Abstract Mathematics
Main Exam (2008), University of Edinburgh
[3] MATH08004, Group Theory: an Introduction to Abstract Mathematics
Main Exam (2011), University of Edinburgh
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