Question

Multiply $137592$ by the smallest number so that the product is a perfect cube.

Answer 1

Prime factorising $137592$, we get,

$137592=2\times 2\times 2\times 3\times 3\times 3\times 7\times 7\times 13$

$=2^3\times 3^3\times 7^2\times 13$.

We know, a perfect cube has multiples of $3$ as powers of prime factors.

Here, number of $2$'s is $3$, number of $3$'s is $3$, number of $7$'s is $2$ and number of $13$'s is $1$.

So we need to multiply another $7$ and ${13}^2$ to the factorization to make $137592$ a perfect cube.

Hence, the smallest number by which $137592$ must be multiplied to obtain a perfect cube is $7\times {13}^2=1183$.