richs_ca wrote:Not sure why I'm finding this confusing, but ....
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 lowest common multiple of x and y is 100.
first, notice that 8 = 2 x 2 x 2, so the prime factorization of 8 is three 2's. this fact will play a key role in all that follows.
(1)
this means that the prime factorization of
each of the numbers x and y contains a 5 and a 2, but that they have no other prime #s in common. (note: this doesn't mean they don't contain other primes - just that there aren't any other primes that are
common to both numbers.)
so:
minimally, xy doesn't have to be a multiple of 8, because the product of x and y only
must contain two 2's (one from x and one from y). for instance, if x and y are both actually 10, then the product, 100 (= 2 x 2 x 5 x 5) is not divisible by 8.
however, xy
can be divisible by 8, if there is another 2 in the prime factorization of either x or y. for instance, if x = 10 (= 2 x 5) and y = 20 (= 2 x 2 x 5), then xy = 200 (= 2 x 2 x 2 x 5 x 5) and is divisible by 8.
so, insufficient.
(2)
this means that 100 (= 2 x 2 x 5 x 5) is the
smallest number of which both x and y are factors.
inter alia, this means that
at least one of x and y must contain two 2's, and
at least one of them must contain two 5's.
but that doesn't answer the question of whether there are three 2's in the product. consider the following:
* if x = 4 (= 2 x 2) and y = 25 (= 5 x 5), then the lcm is 100, but the product (which is also 100) is not divisible by 8.
* if x = 4 (= 2 x 2) and y = 50 (= 2 x 5 x 5), then the lcm is still 100, but the product IS divisible by 8.
so, insufficient
(together)
as stated in (1), each of x and y must contain at least one 2.
as stated in (2), one of them must contain two 2's.
so, the product contains at least one plus two = three 2's.
so it's divisible by 8
sufficient.
answer = c
--
here's a
really cool fact about gcf and lcm:
(gcf of x and y) x (lcm of x and y) = x times y
this is
always true. it's badass.
notice its application to this problem: if you take statements (1) and (2) together, you immediately have the result that the product of x and y is 10 x 100 = 1000, which is divisible by 8.
sweetness
