A Square and A Cube

Perfect squares and cubes, their patterns, finding square and cube roots by prime factorisation, and estimating roots.

Key ideas

A perfect square ends in 0, 1, 4, 5, 6 or 9, and has an even number of zeros at the end. A number ending in 2, 3, 7 or 8 is never a square.
The sum of the first n odd numbers is n²: 1 + 3 + 5 + 7 = 16 = 4².
Square root by prime factorisation: pair up equal prime factors and take one from each pair.
Cube root: group equal prime factors in threes and take one from each group.
To make a perfect square (or cube), multiply by the primes needed to complete every pair (or triple).
Estimate a root between two known squares: 40 lies between 36 and 49, so √40 is between 6 and 7.
1729 = 1³ + 12³ = 9³ + 10³, the smallest number that is a sum of two cubes in two ways (the Hardy–Ramanujan number).

Thinking every number ending in 1, 4, 5, 6 or 9 is a square (11 and 14 are not).
Taking one factor from each triple when finding a square root (use pairs for squares).
Saying a negative number has no cube root.

Find the smallest number by which 72 must be multiplied to make a perfect cube.

Working: 72 = 2³ × 3²: make every power a multiple of 3 → × 3 = 216 = 6³

Answer: 3