Theorem

A perfect square has every prime exponent even

Arithmetic • Number System • Perfect Squares

Formula

N = \prod p_{i}^{a_{i}} is a square ⟺ every a_{i} is even

A positive integer

N = p_{1}^{a_{1}} p_{2}^{a_{2}} \dots p_{k}^{a_{k}}

is a perfect square if and only if every exponent

a_{1}, a_{2}, \dots, a_{k}

is even. The least multiplier that turns

N

into a perfect square is therefore the product of the primes whose exponent in

N

is odd.

How to use it

1To decide whether a number is a square, factorise it and look only at the parity of the exponents.
2To find the least number to multiply (or divide) by to reach a perfect square, collect exactly the primes appearing to an odd power.
3For the least perfect square divisible by several numbers, take their LCM first, then top up its odd exponents.

Worked example

What is the least perfect square divisible by

12

Step-by-step

1
$12 = 2^{2} \cdot 3$
Why: Factorise, then read the exponents: $2$ is even, $1$ is odd.
2
$12 \times 3 = 36 = 2^{2} \cdot 3^{2} = 6^{2}$
Why: Only the prime $3$ sits at an odd exponent, so $3$ is the least multiplier.

Answer

36

Watch out

The criterion is about the exponents, not the primes: multiplying by the whole LCM again is almost always more than is needed, because the primes already at even powers do not have to be touched. It applies to perfect squares only — for a perfect cube the condition is that every exponent is a multiple of

3

Builds on

Prime factorisation

Also known as

perfect square test, every exponent even, square if and only if exponents are even, least multiplier to make a perfect square, prime factorisation of a square

Appears in past papers

2020 (I)

Q47