if a and b are positive even integers, and the least common

airan wrote:If a and b are positive even integers, and the least common multiple of a and b is expressed as ab/n, which of the following statements could be false ?

1. n is a factor of both a and b
2. ab/n < ab

3. ab is multiple of 2.

4. ab/n is a multiple of 2.

5. n is a multiple of 4.

This question is easier if you know that, for any positive integers a and b:

LCM(a,b) * GCD(a,b) = a*b

So LCM(a,b) = ab/GCD(a,b)

In this question, n is just the GCD of a and b. So we know automatically that n is a factor of a and b... A) must be true. If a and b are both even, they are both divisible by 2, so the smallest possible value for their GCD is 2. That is, n >= 2, so B) must be true (I'd add, in response to reachac's comment above, we do not need to have <= here). If a and b are even, ab is clearly even, and so is the LCM of a and b, so C) and D) must be true. E) is the only one that might be false; E will only be true if both a and b are divisible by 4.