Technique

Least perfect square not below a given bound

Arithmetic • Number System • Perfect Squares

Statement

For a positive integer

N

, the least perfect square

\geq N

m^{2}

where

m = ⌈ N ⌉

, and the greatest perfect square

\leq N

k^{2}

where

k = ⌊ N ⌋

. Nothing strictly between

k^{2}

and

(k + 1)^{2}

is a square, because squares of consecutive integers leave no square between them.

How to use it

1Estimate $N$ between two consecutive integers, then square the upper one; squaring the lower one instead answers the opposite question.
2'The smallest $d$ -digit perfect square' means the least square $\geq 1 0^{d - 1}$ — a lower bound to clear, not the number $1 0^{d - 1}$ itself.
3'Least number to be added to $N$ to make a perfect square' is $⌈ N ⌉^{2} - N$ ; 'least to be subtracted' is $N - ⌊ N ⌋^{2}$ .

Worked example

What is the smallest perfect square that is not less than

500

, and how much must be added to

500

to reach it?

Step-by-step

1
$2 2^{2} = 484 < 500$ and $2 3^{2} = 529 \geq 500$
Why: Bracket $500$ between consecutive integers: $500 \approx 22.36$ .
2
$m = ⌈ 500 ⌉ = 23$ , so the square is $529$
Why: Round the square root up, then square.
3
$529 - 500 = 29$

Answer

529

; add

29

Watch out

Take the ceiling of the root, not of the number: rounding

N

up by one place too far gives the SECOND square above

N

, which is the commonest wrong answer. When

N

is itself a square the two bounds coincide and nothing need be added.

Also known as

smallest perfect square greater than, least square above a bound, smallest four-digit perfect square, integer square root then square, least number to add to make a perfect square, nearest perfect square

Least perfect square not below a given bound

Statement

Worked example

Appears in past papers

2020 (I)

2019 (I)

2018 (I)