Question

find the smallest number by which the given number must be multiplied so that the product is a perfect square .1)23805
2)12150
3)7688.​

Answer 1

Step-by-step explanation:

23805

The prime factors for 23805

23805 = 3×3×23×23×5 ( grouping the prime factors in equal pairs we get,)

= (3×3) × (23×23) × 5

Here 5 is left out.

So, multiply by 5 we get, = 23805 × 5 = (3×3) × (23×23) × (5×5)

= (3×5×23) × (3×5×23) = 345 × 345

= (345)2

(ii) 12150 The prime factors for 12150

12150 = 2×2×2×2×3×3×5×5×2

(grouping the prime factors in equal pairs we get,)

= (2×2) × (2×2) × (3×3) × (5×5) × 2

Here, prime factor 2 is left out.

So, multiply by 2 we get, = 12150 × 2 = (2×2) × (2×2) × (3×3) × (5×5) × (2×2)

= (2×2×3×5×2) × (2×2×3×5×2)

= 120 × 120

= (120)2

(iii) 7688 The prime factors for 7688

7688 = 2×2×31×31×2

(grouping the prime factors in equal pairs we get,)

= (2×2) × (31×31) × 2

Here, prime factor 2 is left out. So, multiply by 2 we get,

= 7688 × 2

= (2×2) × (31×31)× (2×2)

= (2×31×2) × (2×31×2)

= 124 × 124

= (124)2