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Problem 655

AIME late

Number theory Difficulty 5.2 Find the answer

## 37. How old is Florence?

Try to determine this yourself, and we will only indicate that the difference between the fifth power of the desired number of years and the number itself is divisible by 10.

This one wants a proof. Work it on paper, read the official solution, then mark yourself honestly — the ladder only means something if the record is true.

Official solution

37. Let $a$ be the age of Florence, which we want to determine. Then

$a^{5}-a=10 k$

where $k$ is some integer. Factoring the left side of the equation, we get

$a\left(a^{2}-1\right)\left(a^{2}+1\right)=10 k$

i.e.

$a(a+1)(a-1)[(a-2)(a+2)+5]=10 k .$

No matter what $a$ is, one of the two values, $a$ or $a+1$ , is divisible by 2, and one of the five values, $a, a+1, a+2$ , $a-1$ or $a-2$ , is divisible by 5. Therefore, $a^{5}-a$ is divisible by 10 for any $a$ . In other words, from the conditions of the problem, we cannot learn anything about Florence's age!

Source: NuminaMath-1.5, licensed Apache-2.0. Statement and solution reproduced as published; topic, difficulty and ordering added by this site.