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Problem 526

AMC 12 late, AIME early

Number theory Difficulty 4.9 Find the answer

The natural number $A$ has exactly 100 different divisors (including 1 and $A$ ). Find their product.

Your answer

Official solution

Each divisor $d$ of the number $A$ corresponds to another divisor $A / d$ .

## Solution

Notice that if the number $d$ is a divisor of the number $A$ , then the number $A / d$ is also a divisor of the number $A$ . Since $A$ has 100 divisors, they are split into 50 such pairs.

## Answer

$A^{50}$

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