CSci/Math2112
Assignment 9
Due July 30, 2015
This assignment is due on Thursday, July 30th at 10am.
(10)
1. (BoP 11.3 #4) Suppose P is a partition of a set A. Define a relation R on A by declaring xRy if and
only if x, y ∈ X for some X ∈ P . Prove R is an equivalence relation on A. Then prove that P is the set
of equivalence classes of R.
Solution: To show that R is an equivalence relation, we have to show that it is reflexive, symmetric,
and transitive.
Reflexive: Fix an x ∈ A. Let X be the part of the partition P such that x ∈ X. Then xRx since
X ∈ P fulfills the requirement for the relation.
Symmetric: Suppose that xRy for some x, y ∈ A. Then by definition of R, there exists an X ∈ P such
that x, y ∈ X. In particular, that also means that y, x ∈ X, thus yRx.
Transitive: Suppose that xRy and yRz for some x, y, z ∈ A. Then by definition, there exist X1 , X2 ∈ P
such that x, y ∈ X1 and y, z ∈ X2 . Since P is a partition of A, y is only in one part, thus X1 = X2 .
Therefore x, z ∈ X1 , which shows xRz.
Thus R is reflexive, symmetric, and transitive, so it is an equivalence relation.
Now let X ∈ P be a part of the partition. By definition of R, we have xRy for any two x, y ∈ X.
Furthermore, given elements a ∈ X and b ∈ A − X, we have by definition that a and b are not related.
Thus X is an equivalence class of R. Finally, since every element of A is in exactly one of the parts of
P , we have that there are no equivalence classes beside the ones from the partition. Thus P is the set
of equivalence classes of R.
(8)
2. Write the addition and multiplication tables for Z7 .
Solution:
+ [0]
[0] [0]
[1] [1]
[2] [2]
[3] [3]
[4] [4]
[5] [5]
[6] [6]
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[1]
[1]
[2]
[3]
[4]
[5]
[6]
[0]
[2]
[2]
[3]
[4]
[5]
[6]
[0]
[1]
[3]
[3]
[4]
[5]
[6]
[0]
[1]
[2]
[4]
[4]
[5]
[6]
[0]
[1]
[2]
[3]
[5]
[5]
[6]
[0]
[1]
[2]
[3]
[4]
[6]
[6]
[0]
[1]
[2]
[3]
[4]
[5]
×
[0]
[1]
[2]
[3]
[4]
[5]
[6]
[0]
[0]
[0]
[0]
[0]
[0]
[0]
[0]
[1]
[0]
[1]
[2]
[3]
[4]
[5]
[6]
[2]
[0]
[2]
[4]
[6]
[1]
[3]
[5]
[3]
[0]
[3]
[6]
[2]
[5]
[1]
[4]
[4]
[0]
[4]
[1]
[5]
[2]
[6]
[3]
[5]
[0]
[5]
[3]
[1]
[6]
[4]
[2]
[6]
[0]
[6]
[5]
[4]
[3]
[2]
[1]
3. Do each of the following calculations in Z7 . Write your result as the equivalence class [a] where 0 ≤ a < 7.
(a) [2] + [4]
(b) [2] · [6]
(c) [5]/[1]
(d) [0]/[3]
(e) [1]/[6]
Solution: [2] + [4] = [6]
[2] · [6] = [5]
[5]/[1] = [5]
[0]/[3] = [0]
[1]/[6] = [6]
(6)
4. Find 8262 mod 60.
Solution: First, 262 = 256 + 4 + 2. We have the following:
81 ≡ 8
(mod 60)
2
8 ≡ 64 ≡ 4
4
2 2
8
4 2
(mod 60)
8 ≡ (8 ) ≡ 42 ≡ 16
(mod 60)
2
8 ≡ (8 ) ≡ 16 ≡ 256 ≡ 16
16
≡ 16 ≡ 16
32
≡ 16
(mod 60)
64
8
≡ 16
(mod 60)
128
≡ 16
(mod 60)
256
≡ 16
(mod 60)
8
8
8
8
2
(mod 60)
(mod 60)
Therefore
8262 = 8256 · 84 · 82
≡ 16 · 16 · 4
≡ 16 · 4
≡4
(4)
(mod 60)
(mod 60)
(mod 60)
5. Find the inverse of 11 modulo 40.
Solution: We have the following calculations using the Division Algorithm:
40 = 3 · 11 + 7
11 = 1 · 7 + 4
7=1·4+3
4=1·3+1
Using these equalities we get:
1=4−3
= 4 − (7 − 4) = 2 · 4 − 7
= 2(11 − 7) − 7 = 2 · 11 − 3 · 7
= 2 · 11 − 3(40 − 3 · 11) = 11 · 11 − 3 · 40
The equality 1 = 11 · 11 − 3 · 40 taken modulo 40 becomes 11 · 11 ≡ 1 (mod 40). Thus the inverse of
11 modulo 40 is 11 mod 40.
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6. Let φ(n) be Euler’s Totient Function. Find φ(4620).
Solution: The prime factorization of 4620 is 4620 = 22 · 3 · 5 · 7 · 11. Using the formula for Euler’s
Totient Function, we then have
Y 1
φ(4620) = 4620
1−
p
p|4620
1
1
1
1
1
= 4620 1 −
1−
1−
1−
1−
2
3
5
7
11
1
2
4
6
10
= 4620
2
3
5
7
11
= 960.
(4)
7. Alice wants to send Bob a message using an RSA code. Bob has made the following public: n = 21 and
the public key e = 5. Alice converts her message into the number 19. What is the encrypted message
she should send Bob?
Solution: The encrypted message is 195 mod 21. Since 5 = 4 + 1 and
191 ≡ 19
2
19 ≡ 4
(mod 21)
(mod 21)
194 ≡ 16
(mod 21)
we have
195 = 194 · 191
≡ 16 · 19
≡ 10
(mod 21)
(mod 21).
Thus the encrypted message that Alice should send Bob is the number 10.