Page 15 - NCERT Books

SQUARES AND SQUARE ROOTS 103

8. 2025 plants are to be planted in a garden in such a way that each row contains as
many plants as the number of rows. Find the number of rows and the number of
plants in each row.
9. Find the smallest square number that is divisible by each of the numbers 4, 9 and 10.
10. Find the smallest square number that is divisible by each of the numbers 8, 15 and 20.

6.5.4 Finding square root by division method
When the numbers are large, even the method of finding square root by prime factorisation
becomes lengthy and difficult. To overcome this problem we use Long Division Method.

For this we need to determine the number of digits in the square root.
See the following table:
Number Square
10 100 which is the smallest 3-digit perfect square
31 961 which is the greatest 3-digit perfect square

32 1024 which is the smallest 4-digit perfect square
99 9801 which is the greatest 4-digit perfect square
So, what can we say about the number of digits in the square root if a perfect
square is a 3-digit or a 4-digit number? We can say that, if a perfect square is a
3-digit or a 4-digit number, then its square root will have 2-digits.
Can you tell the number of digits in the square root of a 5-digit or a 6-digit
perfect square?

The smallest 3-digit perfect square number is 100 which is the square of 10 and the
greatest 3-digit perfect square number is 961 which is the square of 31. The smallest
4-digit square number is 1024 which is the square of 32 and the greatest 4-digit number is
9801 which is the square of 99.

THINK, DISCUSS AND WRITE
n
Can we say that if a perfect square is of n-digits, then its square root will have
(n + 1) 2
digits if n is even or if n is odd?
2

The use of the number of digits in square root of a number is useful in the following method:
• Consider the following steps to find the square root of 529.
Can you estimate the number of digits in the square root of this number?
Step 1 Place a bar over every pair of digits starting from the digit at one’s place. If the
number of digits in it is odd, then the left-most single digit too will have a bar.
Thus we have, 529 . 2

Step 2 Find the largest number whose square is less than or equal to the number under the 2 529
2
2
extreme left bar (2 < 5 < 3 ). Take this number as the divisor and the quotient – 4
with the number under the extreme left bar as the dividend (here 5). Divide and
1
get the remainder (1 in this case).