The "Interesting Integer"
First look at the clues.
The only digit that is a square and a cube is 1, so the
digit in the hundred thousands place is 1.
The digit in the ones place must be either five or
zero.
The digit in the hundreds place is a cube. Only 1 and
8 are cubes less than 10. One has already been used.
Eight is greater than 5 so the digit in the hundreds
place is 8. This leads to the digit in the ones place
being 5.
The tens digit must be four. According to the first
clue, the tens digit is square. Three square numbers
are less than ten: 1, 4 and 9. We can rule out 9
because it is greater than 5 in the ones place. We can
Tule out one because no digit occurs more than once,
and the digit in the hundred thousands place is one.
The integer so far is 1_ _,845. The only digits left
that are less than five are 0, 2, and 3. We know the
integer is divisible by three so we can use the
divisibility rule for 3. The sum of 1, 8, 4 and 5 is 18,
a multiple of three. Thus, the sum of the other two
digits must be a multiple of three as well. The sum of
o and 3 equals 3 which is a multiple of three.
So our number is either 103,845, or 130,845. 130,835
divided by 7 equals 18,692.142857
103,845 by 7
equals 14,835.
Thus, the number I'm thinking of is 103,845.
Bonus:
Dividing 103,845 by 3x5x7 == 989.
Dividing 989 by prime numbers results in 23 and 43
as factors.
Thus, the prime factors of 103,845 are 3,5,7,23
43.
and
Sample Solutions for Interesting
Integer
Sample 1) The "Interesting Integer" is 103,845.
Let's denote the "Interesting Integer" by N.
We make the following deductions
about N.
<Claim 1>
The units (ones) digit of N is 5.
<Why?>
The units digit has to be 0 or 5, because N is divisible by 5. But the units digit
cannot be 0, because of the given fact that all digits except the hundreds digit are
smaller than the units digit.
<Claim 2>
The hundreds digit is 8.
<Why?>
The hundreds digit must be 0, 1 or 8, because we know that it is a cube. But we also
know that the hundreds
digit is bigger than the units digit, which is 5. Out of 0, 1, or
8, the only one that is
bigger than 5 is 8.
<Claim 3>
N has 6 or 7 digits.
<Why?>
The hundred thousands digit appears in the problem, so there are at least six digits. On
the other hand, we know that with the exception of 8, all the remaining digits must be
smaller than 5, and that they
are all distinct. So we have a 5 and an 8 and at most one of each of 0, 1, 2, 3, 4; this
gives a total of at most 7 digits.
<Claim 4>
N has six digits.
<Why?>
Suppose that N had 7 digits. As we saw in the argument for Claim 3, the 7 digits of N
must be: 0, 1, 2, 3, 4, 5, and 8 (in some order). But then the sum of the digits would
be 23, which is not divisible by 3.
This contradicts the given fact that N is divisible by 3. (We used here the test for
divisibility by 3, which says that N is divisible by
3 if and only if so is the sum of its digits.)
<Claim 5>
The hundred thousands digit of N is a 1.
<Why?>
This digit is both a cube and a square so it must be 0 or 1. But it cannot be 0 because
it is the first digit of the number N.
From the Claims 1-5 we deduce that N = lxy8z5, where x, y, and z are distinct digits
from the set {O, 2, 3, 4). We have extra information about z: we know that z is a
square, so
z = 0 or z = 4.
Case 1: z=O.
In this case, x and yare two distinct digits from {2, 3, 4).
The sum of digits of N is l+x+y+8+0+5 = x+y+14.
The sum of the digits of N has to be divisible by 3 (because N is divisible by 3 and by
using the test for divisibility by 3). But this can only be if x=3 and y=4, or x=4 and
y=3.
So in this case, N is 134805 or 143805. But none of these numbers is divisible by 7. So
-- No solution in Case I!
Case 2: z=4.
In this case, x and yare two distinct digits from {2, 3, O}.
The sum of digits of N is 1+x+y+8+4+5 = x+y+18.
The sum of the digits of N has to be divisible by 3 (because N is divisible by 3 and by
the test for divisibility by 3). But this can only be if x=3 and
y=O, or x=O and y=3.
So in this case, N is 130845 or 103845. Out of these numbers, only the second one is
divisible by 7.
So the only possible solution is N = 103845. We see that indeed this number satisfies
all the conditions given in the problem.
BONUS:
In order to obtain the prime factorization of the "Interesting Integer" N, we first
divide N by 3, 5, and 7 (we know this will work, from the hypothesis of the problem). We
get: N = 3*5*7*989.
Now it remains to find the prime factorization of 989. We try to divide 989 by prime
numbers up to 31 (we go up to 31 because the square root of 989, rounded to two
decimals, is 31.45). We find 23 is a divisor, and that 989 = 23*43 (both prime numbers).
Hence
N = 3*5*7*23*43.
Sample 2) The "Interesting
First
Integer"
look at the clues.
The only digit that is a square and a cube is 1, so the digit in the hundred thousands
place is 1.
The digit in the ones place must be either five or zero.
The digit in the hundreds place is a cube. Only 1 and 8 are cubes less than 10. One has
already been used. Eight is greater than 5 so the digit in the hundreds place is 8. This
leads to the digit in the ones place being 5.
The tens digit must be four. According to the first clue, the tens digit is square.
Three square numbers are less than ten: 1, 4 and 9. We can rule out 9 because it is
greater than 5 in the ones place. We can rule out one because no digit occurs more than
once, and the digit in the hundred thousands place is one.
The integer so far is 1
, 845. The only digits left that are less than five are 0, 2,
and 3. We know the integer is divisible by three so we can use the divisibility rule for
3. The sum of 1, 8, 4 and 5 is 18, a multiple of three. Thus, the sum of the other two
digits must be a multiple of three as well. The sum of 0 and 3 equals 3 which is a
multiple of three.
So our number is either 103,845,
103,845 by 7 equals 14,835.
or 130,845.130,835
divided
by 7 equals
Thus, the number I'm thinking of is 103,845.
Bonus:
Dividing 103,845 by 3x5x7 = 989.
Dividing 989 by prime numbers results in 23 and 43 as factors.
Thus, the prime factors of 103,845 are 3, 5, 7, 23 and 43.
Sample 3) The answer is 103,845. I used guess and check.
Bonus: 3,5,7,23,43.
I used guess and check.
18,692.142857....