Question

Find the least number which must be added to 252 such that the number becomes a perfect square.

Answer 1

Hint: The number resulted when a number is multiplied by itself is called a square of a number. In general, it is the product of the number by itself. If the given numbers are not the perfect squared number we approach for the numbers which are nearest to it. To check whether the number is a perfect square we factorize the number and their same prime numbers are paired in case of square. Then we find their differences.

Complete step by step solution:
Let the number which is to be added be \[x\]
First check whether the given number is a perfect square number or not, to check we find their prime factors
\[ 2\underline {\left| {252} \right.} \\
   2\underline {\left| {126} \right.} \\
   7\underline {\left| {63} \right.} \\
   3\underline {\left| 9 \right.} \\
   3 \\ \]
Hence the factors are \[\left( {252} \right) = 2 \times 2 \times 3 \times 3 \times 7\]
Now find the pairs of the same factors: \[\left( {252} \right) = \underline {2 \times 2} \times \underline {3 \times 3} \times 7\]
Hence, we can see that all prime factors do not make a pair hence we can say the number is not a perfect square. Now check the for the perfect squared numbers which are near 252,
We know \[{\left( {12} \right)^2} = 144,{\left( {13} \right)^2} = 169,{\left( {14} \right)^2} = 196,{\left( {15} \right)^2} = 225,{\left( {16} \right)^2} = 256,{\left( {17} \right)^2} = 289\]
Hence we can say that 256 is the nearest perfect square number which next lies to the 252
Therefore the value of \[x\] will be equal to
\[ x = 256 - 252 \\
   = 4 \\ \]
Hence, when 4 is added to the number it becomes a perfect square number.

Note: If we are asked to add a number to make a number perfect square or perfect cube first we find the number which is a perfect square or perfect cube respectively nearest to it.