Question

what is the smallest number by which 405*392 is to be multiplied so that the product is a perfect square.
Please answer in step by step process​

Answer 1

Answer:

10

Step-by-step explanation:

The prime factorizations of 405 and 392 are:

405 = 3⁴ × 5¹

392 = 2³ × 7²

The prime factorization of their product is then:

405 × 392 = 2³ × 3⁴ × 5¹ × 7²

To have a square number, each prime factor needs to have an even power.  The 3 already has an even power (power 4), and so does the 7 (power 2).

But the 2 and the 5 have odd powers.  To get a number where these have even powers, the least we can do is multiply by one more 2 and one more 5.  So the smallest multiplier that results in a perfect square is 2 × 5 = 10.

The resulting number is then

10 × 405 × 392

= 2⁴ × 3⁴ × 5² × 7²

= ( 2² × 3² × 5 × 7 )²

= 1260²