Question

Find the smallest number by which $1323$ must be multiplied so that the product is a perfect cube.

Answer 1

Prime factorising $1323$, we get,

$1323=3\times 3\times 3\times 7\times 7$

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

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

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

So we need to multiply another $7$ in the factorization to make $1323$ a perfect cube.

Hence, the smallest number by which $1323$ must be multiplied to obtain a perfect cube is $7$.