Question

fine the least number which must be added to the following numbers to make them perfect squares 4515600​

Answer 1

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

Given number is 4515600.

Since, we have to find the least number that must be added to 4515600 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,

$\begin{gathered} \:\: \begin{array}{c|c} {\underline{\sf{}}}&{\underline{\sf{\:\:2125 \:\:}}}\\ {\underline{\sf{2}}}& {\sf{\:\:4515600 \:\:}} \\{\sf{}}& \underline{\sf{\:\: \: 4 \:  \:  \:  \:  \:  \:  \:\:}} \\ {\underline{\sf{41}}}& {\sf{\:\: \: \: \: 051\:  \:  \:  \:\:\:}} \\{\sf{}}& \underline{\sf{\:\: \: \: \: 41 \:    \:  \: \:\:}} \\ {\underline{\sf{422}}}& {\sf{\:\: \: \: 1056  \:\:}} \\{\sf{}}& \underline{\sf{\: \: \: \: \:844\:\:}} \\ {\underline{\sf{4245}}}& {\sf{\:\: \:  \: \:  \: 21200\:\:}} \\{\sf{}}& \underline{\sf{\:\: \:  \:  \: \: \: 21225\:\:}} \\ {\underline{\sf{}}}& {\sf{\: \:  \:  \:  \:  \:  \:  \:25\:\:}}{\sf{}}&{\sf{\:\:\:\:}}\end{array}\end{gathered}$

So, it means 25 must be added to 361298 to make it a perfect square.

Thus, Required number is 4515600 + 25 = 4515625.

So,

$\begin{gathered} \:\: \begin{array}{c|c} {\underline{\sf{}}}&{\underline{\sf{\:\:2125 \:\:}}}\\ {\underline{\sf{2}}}& {\sf{\:\:4515625 \:\:}} \\{\sf{}}& \underline{\sf{\:\: \: 4 \:  \:  \:  \:  \:  \:  \:\:}} \\ {\underline{\sf{41}}}& {\sf{\:\: \: \: \: 051\:  \:  \:  \:\:\:}} \\{\sf{}}& \underline{\sf{\:\: \: \: \: 41 \:    \:  \: \:\:}} \\ {\underline{\sf{422}}}& {\sf{\:\: \: \: 1056  \:\:}} \\{\sf{}}& \underline{\sf{\: \: \: \: \:844\:\:}} \\ {\underline{\sf{4245}}}& {\sf{\:\: \:  \: \:  \: 21225\:\:}} \\{\sf{}}& \underline{\sf{\:\: \:  \:  \: \: \: 21225\:\:}} \\ {\underline{\sf{}}}& {\sf{\: \:  \:  \:  \:  \:  \:  \:00\:\:}}{\sf{}}&{\sf{\:\:\:\:}}\end{array}\end{gathered}$

Thus,

$\rm \implies\:\boxed{ \tt{ \: \sqrt{4515625} = 2125 \: }}$