Question

Find the least numbers which must be subtracted from the following numbers so as to leave a perfect square:
(i) $2361$ (ii) $4931$ (iii) $18265$ (iv) $390700$

Answer 1

(i) $2361<2500$
$\therefore \sqrt{2361}<50$
$1600<2361$
$\therefore 40<\sqrt{2361}$
$\Rightarrow 40<\sqrt{2361}<50$
$2025<2361$
$\Rightarrow 45<\sqrt{2361}$
$2116<2361$
$\Rightarrow 46<\sqrt{2361}$
$2209<2361$
$\Rightarrow 47<\sqrt{2361}$
$2304<2361$
$\Rightarrow 48<\sqrt{2361}$
$\therefore 2401>2361$
$\Rightarrow 48<\sqrt{2361}<49$
$\Rightarrow$ The nearest square number lesser than $2361$ is $2304$.
$\Rightarrow$ The number to be subtracted from $2361$ to make it a perfect square is $2361-2304=57$.
$\therefore 57$ is the least number to be subtracted from $2361$ to make it a perfect square.
(ii) $4931$
$4900<4931$
$\Rightarrow 70<\sqrt{4931}$
$5041>4931$
$\Rightarrow 71>\sqrt{4931}$
$\Rightarrow$ The nearest square number lesser than $4931$ is $4900$.
$\Rightarrow$ The number to be subtracted from $4931$ to make it a perfect square is $4931-4900=31$.
$\therefore 31$ is the least number to be subtracted from $4931$ to make it a perfect square.
(iii) $18265$
$16900<18265$
$\Rightarrow 130<\sqrt{18265}$
$17161<18265$
$\Rightarrow 131<\sqrt{18265}$
$17424<18265$
$\Rightarrow 132<\sqrt{18265}$
$17689<18265$
$\Rightarrow 133<\sqrt{18265}$
$17956<18265$
$\Rightarrow 134<\sqrt{18265}$
$18225<18265$
$\Rightarrow 135<\sqrt{18265}$
$18496<18265$
$\Rightarrow 136>\sqrt{18265}$
$\Rightarrow$ The nearest square number lesser than $18265$ is $18225$.
$\Rightarrow$ The number to be subtracted from $18225$ to make it a perfect square is $18265-18225=40$.
$\therefore 40$ is the least number to be subtracted from $18265$ to make it a perfect square.
(iv) $390700$
$390625<390700$
$\Rightarrow 625<\sqrt{390700}$
$391876<390700$
$\Rightarrow 625>\sqrt{390700}$
$\Rightarrow$ The nearest square which is lesser than $390700$ is $390625$.
$\Rightarrow$ The number to be subtracted from $390700$ to make it a perfect square is $390700-390625=75$.
$\therefore 75$ is the least number to be subtracted from $390700$ to make it a perfect square.