Question

What is the least number that should be added to each of the following to make these perfect squares?
 $ \left( i \right){\text{ 3169}} $
 $ \left( {ii} \right){\text{ 2765}} $
 $ \left( {iii} \right){\text{ 36,320}} $
 $ \left( {iv} \right){\text{ 81,180}} $
 $ \left( v \right){\text{ 92,700}} $

Answer 1

Hint:
Here the approach we will use in this question to find the least number to make this perfect square is that we will first find the number which is the greatest perfect number near to the question and then we will subtract it to find the least number.

Complete step by step solution:
  $ \left( i \right){\text{ 3169}} $
So we will find the closest square number near to $ {\text{3169}} $ is $ 3249 $ , so the difference between two of them will be given as-
 $ \Rightarrow 3249 - 3169 = 80 $
Therefore, we can say that $ 80 $ should be the least number which when gets added to $ {\text{3169}} $ will make it a perfect square.
Hence, $ \sqrt {3249} = 57 $
 $ \left( {ii} \right){\text{ 2765}} $
So we will find the closest square number near to $ 2765 $ is $ 2809 $ , so the difference between two of them will be given as-
 $ \Rightarrow 2809 - 2765 = 44 $
Therefore, we can say that $ 44 $ should be the least number which when gets added to $ 2765 $ will make it a perfect square.
Hence, $ \sqrt {2809} = 53 $
 $ \left( {iii} \right){\text{ 36,320}} $
So we will find the closest square number near to $ 36,320 $ is $ 36,481 $ , so the difference between two of them will be given as-
 $ \Rightarrow 36,481 - 36,320 = 161 $
Therefore, we can say that $ 161 $ should be the least number which when gets added to $ 36,320 $ will make it a perfect square.
Hence, $ \sqrt {36,481} = 191 $
 $ \left( {iv} \right){\text{ 81,180}} $
So we will find the closest square number near to $ 81,180 $ is $ 81,225 $ , so the difference between two of them will be given as-
 $ \Rightarrow 81,225 - 81,180 = 45 $
Therefore, we can say that $ 45 $ should be the least number which when added to $ 81,180 $ will make it a perfect square.
Hence, $ \sqrt {81,225} = 285 $
 $ \left( v \right){\text{ 92,700}} $
So we will find the closest square number near to $ 92,700 $ is $ 93,025 $ , so the difference between two of them will be given as-
 $ \Rightarrow 93,025 - 92,700 = 325 $
Therefore, we can say that $ 325 $ should be the least number which when gets added to $ 92,700 $ will make it a perfect square.

Hence, $ \sqrt {93,025} = 305 $

Note:
This type of question can also be solved by using the long division method and we can know the least number from there. To solve the above method we have to use the heat and trial method to check the perfect square. I would suggest always checking the number which can be easily squared from there, checking the number around and in this way we can easily find out without much more calculations.