2004 Indonesia MO Problems/Problem 1

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Problem

How many odd and even divisors of $5^6 - 1$ are there?

Solution

We start by factoring the expression. \[(5^3 + 1)(5^3 - 1)\] \[126 \cdot 124\] \[2 \cdot 63 \cdot 4 \cdot 31\] \[2^3 \cdot 3^2 \cdot 7 \cdot 31\] If a divisor is even, then the exponent of the $2$ of the divisor’s prime factorization is at least $1$, so there are $3 \cdot 3 \cdot 2 \cdot 2 = \boxed{36}$ even divisors.

If a divisor is odd, then the exponent of the $2$ of the divisor’s prime factorization is $0$, so there are $3 \cdot 2 \cdot 2 = \boxed{12}$ odd divisors.

See Also

2004 Indonesia MO (Problems)
Preceded by
First Problem
1 2 3 4 5 6 7 8 Followed by
Problem 2
All Indonesia MO Problems and Solutions