Question

Divide $259875$ by the smallest number so that quotient is a perfect square.

Answer 1

Prime factorising, $259875=(3\times 3\times 3)\times (5\times 5\times 5)\times 7\times 11$.

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

Here, the prime factor $7$ and $11$ does not appear in triplet form.

Therefore, $259875$ is not a perfect cube.

Since in the factorization, $7$ and $11$ appears only one time, we must divide the number $259875$ by $7\times 11=77$, then the quotient is a perfect cube.

$\therefore 259875÷77=3375$

$=15\times 15\times 15={15}^3$, which is a perfect cube.

$\therefore$ The smallest number by which $259875$ should be divided to make it a perfect cube is $77$.