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Example 23: Find the least number by which 1058 be divided to make it a perfect square. Also,
find the square root of the perfect square so obtained.

Solution: Let us find the prime factors of 1058. 2 1058
23 529
1058 = 23 × 23 × 2 23 23
1
Here, factor 2 does not exist in a pair, 1058 is clearly not a perfect square.
Hence, 2 is the smallest number by which 1058 be divided to make it a perfect square.

1058
⇒ = 529
2
And 529 = 23 × 23 = 23 2

Thus, 529 = 23
Example 24: Find the diagonal of a rectangular field whose length is 20 m and breadth is 15 m.

Solution: Length of the field (BC) = 20 m 20 m
B C
Breadth of the field (AB) = 15 m
From Pythagoras theorem, in right-angled DABC,

AC = AB + BC 2 15 m
2
2
2
⇒ AC = (AB ) + (BC ) 2
A D
= (15 ) + (20 ) 2
2
5
= 225 + 400 = 625 = 5 ××× 5
5
= 25 m
Hence, the diagonal of the rectangular field is 25 m.

Practice Time 5C

1. Using prime factorisation method, find the square root of following.
(a) 5625 (b) 7056 (c) 2601 (d) 1024
2. Find the least number by which the following numbers should be multiplied to obtain a perfect
square. Also find the square root of the perfect square so obtained.
(a) 768 (b) 1575 (c) 2028 (d) 3200

3. Find the least number by which the following numbers should be divided to obtain a perfect square.
Also find the square root of the perfect square so obtained.

(a) 396 (b) 2645 (c) 6480 (d) 8664
4. Find the dimensions of a square field whose area is 7056 square metres.

Finding Square Root by Division Method

When the numbers given are large even the prime factorisation method is tedious and time
consuming. So, we use long division method. To find the square root of a number by division
method, follow the steps as explained.

137 Squares and Square Roots