Question

Find the smallest number that must be added to get a perfect square.
(i) $2361$ (ii) $4931$ (iii) $18265$ (iv) $390700$

Answer 1

(i) $2361$
$2500>2361$
$\Rightarrow 50>\sqrt{2361}$
$2401>2361$
$\Rightarrow 49>\sqrt{2361}$
$2304<2361$
$\Rightarrow 48<\sqrt{2361}$
$\therefore 2401$ is the perfect square nearest to $2361$.
$\therefore 2401-2361=40$ is the number to be added.
$\therefore 40$ is the least number to be added to $2361$ to make it a perfect square.
(ii) $4931$
$4900<4931$
$\Rightarrow 70<\sqrt{4931}$
$5041>4931$
$\Rightarrow 71>\sqrt{4931}$
$\Rightarrow 5041$ is the perfect square nearest to $4931$.
$\Rightarrow 5041-4931=110$ is the number to be added.
$\therefore 110$ is the least number to be added to $4931$ to make it a perfect square.
(iii) $18265$
$18225<18265$
$\Rightarrow 135<\sqrt{18265}$
$18496>18265$
$\Rightarrow 136>\sqrt{18265}$
$\Rightarrow 18496$ is the perfect square nearest to $18265$.
$\Rightarrow 18496-18265=231$ is the number to be added to $18265$.
$\therefore 231$ is the least number to be added to $18265$ to make it a perfect square.
(iv) $390700$
$390625<390700$
$\Rightarrow 625<\sqrt{390700}$
$391876>390700$
$\Rightarrow 626>\sqrt{390700}$
$\Rightarrow 391876$ is the perfect square nearest to $390700$.
$\Rightarrow 391876-390700=1176$ is the number to be added to $390700$.
$\therefore 1176$ is the least number to be added to $390700$ to make it a perfect square.