Question

A General wishes to draw up his 7500 soldiers in the form of a largest square. After arranging, he found out that some of them are left out. How many soldiers were left out?

Answer 1

Hint: Firstly, we will apply approximation and find the number whose square is closest to the number of total soldiers which is 7500. Then, we will subtract the squared number value from the total number of the soldiers. Then, on subtraction we get the number of the soldiers who are left out to form a largest square.

Complete step-by-step answer:
In this question, we have to go by estimation as it requires a number which is a perfect square closer to the value 7500.
Here, the number of the soldiers is 7500 and they need to be arranged in the form of the largest square.
It is clearly evident by seeing that it is less than the square of 90 as the square of 90 is:
${{90}^{2}}=8100$
Now, by applying approximation in the question and choosing the number smaller than 90 to get its perfect square.
So, we choose the number as 85 and find its square as:
${{85}^{2}}=7225$
The square of 85 is less than 7500 then we should move to the next number which is 86.
Find the square of 86 as:
${{86}^{2}}=7396$
Again, the square of 86 is smaller than 7500 then we should move to the next number which is 87
Find the square of 87 as:
${{87}^{2}}=7569$
The square of the number 87 gives the value larger than 7500. So, it is clear that we need to subtract the square of the number 86 from the value 7500 to get the remaining soldiers to form a largest square.
Now, to get the number of the soldiers left is:
$7500-7396=104$
Hence, 104 soldiers are left out.

Note: Another approach to solve this type of the problem is:
Firstly we can find the square root of the total number of the soldiers given which automatically gives us the closest approximation of the nearest natural number whose complete square is possible.
So, by finding the square root of 7500 as:
$\sqrt{7500}=86.6$
It is clear from the above calculation that the number which makes the largest square from the given number of soldiers is 86.
 Now, to get the number of the soldiers left is:
$\begin{align}
  & 7500-{{86}^{2}}=7500-7396 \\
 & \Rightarrow 104 \\
\end{align}$
Hence, 104 soldiers are left out.