Question

What least number must be added to 6072 to make it a perfect square?
a. 6
b. 10
c. 12
d. 16

Answer 1

Hint: We will use the long division method to find the nearest perfect square before and after the given number 6072. In the final step, we will subtract the number 6072 from the greatest perfect square among two, to get the required number.

Complete step-by-step answer:
It is given in the question that we have to find the least number that must be added to 6072 to make it a perfect square. We will use the long division method to find the nearest perfect square just before and after 6072. We can find the square root of 6072 by using the following steps.
Step I: We will first group the digits in pairs, starting with the digit in the units place.
Step II: We will then think of the largest number whose square is equal to or just less than the first period. We will take this number as the divisor and also as the quotient.
Step III: We will subtract the product of the divisor and the quotient from the first period and bring down the next period to the right of the remainder. This becomes the new dividend.
Step IV: We will obtain our new divisor by multiplying the first divisor with two and then again thinking of the largest number which will divide our next dividend.
Step V: We will repeat steps (II), (III) and (IV) till all the periods have been taken up.
The obtained quotient will be our required root of that number.

Thus, we get 143 as the remainder and 77 as the nearest perfect square before 6072.
We know that the square of 77 is 5929 and the next number of 77 is 78, whose square is 6084. So, we can write,
$\begin{align}
  & {{\left( 77 \right)}^{2}}<6072<{{\left( 78 \right)}^{2}} \\
 & 5929<6072<6084 \\
\end{align}$
But, it is given in the question that we have to add some number to 6072 to make it a perfect square.
Thus, we will subtract 6072 from 6084 to get the required number, so we will get,
(6084 – 6072) = 12
Thus, when we add 12 to 6072, we get 6084, which is a perfect square of 78. Therefore the required number is 12.

So, the correct answer is “Option (c)”.

Note: Most of the students make mistakes in the initial step, where they may take the number 77, and take its square, that is 5929 and subtract it from the number 6072. But, it is wrong as we have already obtained the remainder as 143 while dividing the numbers. So, students must find perfect squares on either side of the given number and compare the difference of the number on the higher side to get the least integer to be added to it.