The Three Conditions

Number Theory Level 3

Determine the smallest positive integer $n$ such that:

(1) $n$ is divisible by 20,

(2) $n^{2}$ is a perfect cube, and

(3) $n^{3}$ is a perfect square.

The answer is 1000000.

1 solution

Otto Bretscher
May 3, 2015

By condition (1), we must have $n=2^p5^qm$ , where $p>1$ and neither 2 nor 5 divides $m$ . By condition (2), $p$ and $q$ must both be divisible by 3, and by condition (3) they must be divisible by 2. Thus the smallest such number is $n=2^65^6=\boxed{1000000}$ .

The Three Conditions

The answer is 1000000.

1 solution

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