didieravoaka wrote:Does anyone can help me figure this solution out?
I know that N has to be a multiple of 30. But how come that we have this answer?
Thanks.

It's not that N itself is a multiple of 30, but rather, that each base in N must be raised to an exponent that is a multiple of 30. So the question is, what do we need to multiply a^3 * b^4 * c^5 by so that each base is raised to a multiple of 30? If we multiply a^3 * b^4 * c^5 by a^27 * b^26 * c^25 we get a^30 * b^30 * c^30, so each base is, in fact, raised to a multiple of 30.

The basic idea here:

To be a perfect square, each exponent must be even.

To be a perfect cube, each exponent must be a multiple of 3.

To be a perfect fifth power, each exponent must be a multiple of 5.

So we want each exponent to be divisible by 2, 3, and 5, i.e. by the LCM of 2, 3, and 5, or 30.

So if we have

(abc)³�

we can write it as

((abc)¹�)²

((abc)¹�)³

((abc)�)�

and see that it's a square, a cube, and a fifth.

From that point, we just need the missing powers to make (abc) raised to the 30th, and we're done!

Thanks guys for the explanation.

didieravoaka wrote:Thanks guys for the explanation.

No prob. This is one of the ten or fifteen most frequently asked questions from our practice tests: I feel like I explain it at least once a week to our students!

By the way, I really like your tests. I did the second test today and got 450. I got 470 on the first one. My school asks me to get 450. I plan to do all the 6 tests and retake them once I'll be done.