Question

Find the least number which must be added to each of the following numbers so as to get a perfect square. Also, find the square root of the perfect square so obtained.
$525$
$1750$

• A. $4,23;14,48$
• B. $4,27;14,42$
• C. $4,23;14,42$
• D. $4,27;14,48$

Answer 1

The correct option is C $4,23;14,42$

We try to find out the square root of $525$.
$22\overline{2\mid \overline{5}\overline{25}}\mid -4\overline{42\mid 125}\underset{–}{\mid -84}\mid 41$
We observe that
$(22{)}^2<525<(23{)}^2$
The required number to be added
$=(23{)}^2-525$
$=529-525=4$
$\therefore$ Required perfect square number
$=525+4=529$
Clearly, $\sqrt{529}=23$.
We try to find out the square root of $1750$
$41\overline{4\mid \overline{17}\overline{50}}\mid -16\overline{81\mid 150}\underset{–}{\mid -81}\mid 69$
We observe that $(41{)}^2<1750<(42{)}^2$.
The required number to be added
$=(42{)}^2-1750$
$=1764-1750$
$=14$
$\therefore$ Required perfect square number
$=(1750+14)=1764$
Clearly, $\sqrt{1764}=42$.