2004 Indonesia MO Problems/Problem 1

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.

2004 Indonesia MO (Problems)
Preceded by First Problem	1 • 2 • 3 • 4 • 5 • 6 • 7 • 8	Followed by Problem 2
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2004 Indonesia MO Problems/Problem 1

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