Properties of numbers

[heshamelaziry]

If positive integer x is a multiple of 6 and positive integer y is a multiple of 14, is xy a multiple of 105 ?

(1) x is a multiple of 9.
(2) y is a multiple of 25


Please try to explain as detailed as possible. This might sound boring, but for me Math is a nightmare and specially factors, multiples, divisors,..... Test on the 26th.

Thanks you,

[asamaverick]

Based on what we have
x = 6p
y = 14q

xy = 6*14*pq = 21*4 *pq
Is xy a multiple of 105?
Since xy is a multiple of 21 all we are looking for is if x or y are multiples of 5.

Since (2) answers this, the answer should be B.

[heshamelaziry]

Based on what we have
x = 6p
y = 14q

xy = 6*14*pq = 21*4 *pq
Is xy a multiple of 105?
Since xy is a multiple of 21 all we are looking for is if x or y are multiples of 5.

Since (2) answers this, the answer should be B.

What do you mean by: for is if x or y are multiples of 5 ?

[Stuart@KaplanGMAT]

If positive integer x is a multiple of 6 and positive integer y is a multiple of 14, is xy a multiple of 105 ?

(1) x is a multiple of 9.
(2) y is a multiple of 25

Integers, multiples... looks like a case for prime numbers!

We know that, at a minimum, x has a 2 and a 3 among it's primes. We also know that, at a minimum, y has a 2 and 7 among it's primes. Therefore, at a minimum,

xy = 2*3*2*7

We want to know if xy is a multiple of 105. Well,

105 = 3*5*7

We know that xy already has a 3 and a 7. To answer the question, we need to know if xy also has a prime factor of 5.

(1) x is a multiple of 9. Doesn't tell us if x is a multiple of 5 (x=18 and x=90 are both possible)... insufficient.

(2) y is a multiple of 25. Well, if y is a multiple of 25, we know that y has 5*5 among it's primes: sufficient!

(2) is sufficient, (1) isn't: choose B.

[heshamelaziry]

IMO B