Question

By what smallest number should 216 be divided so that the quotient is a perfect square? Also, find the square root of the quotient.

Answer 1

Prime factors of $216=2\times 2\times 2\times 3\times 3\times 3$
Grouping the factors into pairs of equal factors, we get
$216=2\times 2\times 2\times 3\times 3\times 3$
We find that there is no prime factor to form a pair with 2 and 3.
Therefore, we must divide the number by 6, so that the quotient becomes a perfect square.
If we divide the given number by 6, then
New number $=\frac{216}{6}=36$
Taking one factor from each, we get square root of new number (quotient)
$=2\times 3=6$