Question

when p is multiplied by 585, the product 585p is a perfect square. find the smallest value of p, where p>0

Answer 1

585 prime factories is 3^2x5x13. To make a perfect square, multiply this number to make each factor into an even power. 5x13 works because the prime factorization is now all to the power of 2. The answer is 65.

Answer 2

The smallest value of p is:

65

Explanation:

When p is multiplied by 585, the product 585p is a perfect square.

We will prime factorize 585 and see what factors do not come in a pair and thus the multiplication of such numbers will give us the smallest value for p.

`585=3* 3* 5* 13`

We could observe that 3 comes in pair

while 5 and 13 do not come in pair.

Hence, the value of p will be:

`p=5* 13\\\\i.e.\\\\p=65`

Hence,

`585p=3* 3* 5* 5* 13* 13\\\\i.e.\\\\585p=(3* 5* 13)^2\\\\i.e.\\\\585p=(195)^2`