sumatitandon wrote:IF N and Y are positive integers and 450Y=N^3, which of the following is a integer

1)Y/ 3*2^2*5

2)Y/3^2*2*5

3)Y/3*2*5^2

This one can be solved:

It almost always helps to find the prime factorization in these question types where we ask whether a certain rational expression is an integer.

450Y=N^3
2*3*3*5*5*Y = N^3
For 2*3*3*5*5*Y to be a cube, we need the number of 2's, 3's and 5's to be divisible by 3.
So, for example, 2*2*2*3*3*3*5*5*5 = (2*3*5)^3
For 2*3*3*5*5*Y to be a cube, it must be the case that Y has at least 2 additional 2's, 1 additional 3 and 1 additional 5.
So, Y = 2*2*3*5*(other possible numbers)
We see that 2*2*3*5 is the smallest possible value for Y.

Of the possible answers, only the first one (Y/ 3*2^2*5) will simplify to be an integer.