Question

Find the smallest number by which the given number must be multiplied so the
product is a perfect square:
(i) 23805
(ii) 12150
(iii) 7688
​

Answer 1

Answer:

We are going to find out prime factors of 12150

12150=2×6075

=2×3×2025

=2×3×3×675

=2×3×3×3×225

=2×3×3×3×3×75

=2×3×3×3×3×3×25

=2×3×3×3×3×3×5×5

By grouping the prime factors for 12150

=(3×3)×(3×3)×(5×5)×2×3

By observation, prime factor 2 and 3 are left out.

So, multiply by 2 and 3 we get,

12150×2×3=(3×3)×(3×3)×(5×5)×(2×2)×(3×3)

=(3×3×3×2×5)×(3×3×3×2×5)

=(270)×(270)

=(270)

2

Hence, the smallest number by which 12150 must be multiplied so that the product is perfect square is 6.