Question

Find the least number which must be added to $ 18,265 $ to obtain a perfect square.

Answer 1

Hint: In this question, First we will find the square root of the given number and then we find the square root of the number which is the nearest possible to that number. The number resulting when a number is multiplied by itself is called a square of the number. So we will first find the square of the number i.e. $ 18,265 $ .

Complete step by step solution:
In this problem,
First, We have to find the square root of
 $ 18,265 $
It can be written as
 $ \sqrt {18265} $
The value of
 $ \sqrt {18265} = 135.148 $
Let us take a number which is closer to this
 $ 135 $
We can see that when we square the number $ 135 $ ,
 $ {(135)^2} = 18225 $ .
So according to this number if we want to make the given number a perfect square we have to subtract the number, but in the question we have to add the number to make it a perfect square.
So now we will find the square of the next nearest number i.e.
 $ 136 $ .
Upon squaring it gives us value
  $ {(136)^2} = 18496 $ .
Now we subtract both these numbers i.e.
 $ 18496 - 18265 = 231 $ .
It means that if we add $ 231 $ to $ 18265 $ , then it will be a perfect square.
Hence the required answer is $ 231 $ .
So, the correct answer is “ $ 231 $ ”.

Note: We should note that to check whether the number is a perfect square, we can factorise the number and their prime numbers are paired in case of square. Then we can find their differences. As for example we can write the factors of $ 18225 $ i.e.
 $ 18225 = 3 \times 3 \times 3 \times 3 \times 3 \times 3 \times 5 \times 5 $
Or it can be written as $ 18225 = {3^6} \times {5^2} $
Here the prime factors are $ 3,5 $ .