Question

Find the smallest number which must be added 4000 to to make it a perfect square. Also, find the square root of the perfect square so obtained.​

Answer 1

$\large\underline{\sf{Solution-}}$

Given number is 4000.

Since, we have to find the least number that must be added to 4000 to make it a perfect square.

So, we use long division to find the remainder that should be added to the given number to make it a perfect square.

Thus, Using Long Division Method

$\green{\begin{gathered} \:\: \begin{array}{c|c} {\underline{\sf{}}}&{\underline{\sf{\:\:64 \:\:}}}\\ {\underline{\sf{6}}}& {\sf{\:\:4000 \:\:}} \\{\sf{}}& \underline{\sf{\:\: \:   36  \:  \:  \: \:\:}} \\ {\underline{\sf{124}}}& {\sf{\:\: \:400 \:  \:\:}} \\{\sf{}}& \underline{\sf{\: 496 \: \:\:}} \\ {\underline{\sf{}}}& {\sf{\:\:96 \:\:}} {\sf{}}&{\sf{\:\:\:\:}}\end{array}\end{gathered}}$

So, it means 96 must be added to 4000 to make it a perfect square.

Thus, Required number is 4000 + 96 = 4096.

So,

$\red{\begin{gathered} \:\: \begin{array}{c|c} {\underline{\sf{}}}&{\underline{\sf{\:\:64 \:\:}}}\\ {\underline{\sf{6}}}& {\sf{\:\:4096 \:\:}} \\{\sf{}}& \underline{\sf{\:\: \:   36  \:  \:  \: \:\:}} \\ {\underline{\sf{124}}}& {\sf{\:\:496 \:  \:\:}} \\{\sf{}}& \underline{\sf{\: 496 \: \:\:}} \\ {\underline{\sf{}}}& {\sf{\:\:00 \:\:}} {\sf{}}&{\sf{\:\:\:\:}}\end{array}\end{gathered}}$

Hence,

$\rm \implies\:\boxed{ \tt{ \: \: \sqrt{4096} \: = \: 64 \: \: }}$