Track / Stage 5 / 95 of 400 #695 of 1964

Problem 695

AIME late

Number theory Difficulty 5.2 Find the answer

3 [Divisibility of numbers. General properties]

It is known that $x=2 a^{5}=5 b^{2}>0$ , where numbers $a$ and $b$ are integers. What is the smallest possible value of $x$ ?

Your answer

A number or a short expression. Fractions can be typed as 3/2, and spacing doesn't matter.

Official solution

The number $x$ is divisible by 2 and 5. $x=2 a^{5}$ , so $a$ is divisible by 5. Since $x=5 b^{2}$ , $x$ is divisible by $2^{2}$ , which means $a$ is also divisible by 2. Assuming $a=2 \cdot 5=10$ , we get $x=200000$ and $b=200$ .

## Answer

200000.

Source: NuminaMath-1.5, licensed Apache-2.0. Statement reproduced verbatim; metadata (topic, difficulty, ordering) added by this project.