Find the smallest number by which 3645 must be divided so that it becomes a perfect square.

The prime factorisation of 3645:

3645 = 3 x 3 x 3 x 3 x 3 x 3 x 5

Grouping the factors into pairs of equal factors, we get:

3645 = (3 x 3) x (3 x 3) x (3 x 3) x 5

The factor, 5 does not have a pair. Therefore, we must divide 3645 by 5 to make a perfect square. The new number is:

(3 x 3) x (3 x 3) x (3 x 3) = 729

Taking one factor from each pair on the LHS, the square root of the new number is 3 x 3 x 3, which is equal to 27.