If x and y are positive integers, is xy a multiple of 8?

[gphil]

Could please somebody explain how to approach to such type of questions? Thanks!

If x and y are positive integers, is xy a multiple of 8?

1) The greatest common divisor of x and y is 10
2) The least common multiple of x and y is 100

The correct answer is c) Both statements together are sufficient, but neither statement alone is sufficient.

[TheChakra]

Could please somebody explain how to approach to such type of questions? Thanks!

If x and y are positive integers, is xy a multiple of 8?

1) The greatest common divisor of x and y is 10
2) The least common multiple of x and y is 100

The correct answer is c) Both statements together are sufficient, but neither statement alone is sufficient.

1. implies that x and y both have only 2 common terms 5 and 2. One of them can actually have 2 more 2s (or either have 1 more 2) to make xy divisible by 8. so if x and y are 10 and 10, GCD is still 10, but not divisible. But if the terms are 10 and 20, xy is divisible by 8. Not sufficient

2. implies that 2x2x5x5 is the LCM. A few of those terms are common between x and y and few of those are not. So you don't know if you at least have 3 2s between x and y. Not sufficient.

1 and 2 combine tells you that only 5 and 2 are common between so you have 2 2s. This also means that 2x5 of LCM is "extra" so you get another 2. You have 3 2s and hence divisible.

C is the answer

[RonPurewal]

you've got a pretty sweet explanation here already, so, instead of improving on it, i'll drop a cool fact on you:

for any two positive integers X and Y,
X x Y = (lcm of X and Y) x (gcf of X and Y)

this is a pretty obscure fact, but it's golden in this problem: if you have statements 1 and 2 together, then you can get right away that
X x Y = 10 x 100 = 1000
which is divisible by 8.

[gphil]

Thanks a lot for great explanations!

[RonPurewal]