Question

If 375y=x2 and x and y are positive integers,

then which of the following must be an integer?
I. y/15
II. y/30
III. (y2)/25

a) I only
b) III only
c) I and II
d) I and III
e) I, II and III

Answer 1

Option d - I and III.

Explanation:

Given : If
`375y=x^2` and x and y are positive integers.

To find : Which of the following must be an integer?

Solution :

As we see all option there is a multiple of y.

So, we factoring the number 375

i.e.
`375=3* 5* 5* 5`

`375=15* 5^2`

`375=15* 25`

In order for 375y to be a perfect square,

The prime factorization of y must contain at least one 3 and one 5.

or y must be a multiple of 15.

If y is a multiple of 15, then
`(y)/(15)` must be an integer.

and
`(y^2)/(25)` must be an integer.

Therefore, I and III will be correct i.e. option d.