mniyer wrote:
If a,b,c, and d are all positive integers, is c divisible by a/d?
(1) b * c is divisible by a
(2) GCF (a,b) = d
I'm not sure about the answer and I don't know the source. Appreciate posts with explanation.
Responding to a pm:
Question: Is c divisible by a/d?
(1) b * c is divisible by a
b*c = a*m (m is an integer)
Doesn't tell us anything about d. We could give d any value to get yes or no as answers. Not sufficient alone.
(2) GCF (a,b) = d
a = d*p
b = d*q (p and q are co-prime integers i.e. they have no common factor except 1)
This doesn't tell us anything about c. We could give c any value to get yes or no as answers. Not sufficient alone.
Now the answer will be either (C) or (E).
Using both statements together, let's get everything in terms of d for comparison.
a = d*p
b = d*q
c = a*m/b = dpm/dq = pm/q
Now, since p and q have no common factors but c is an integer, m must be a multiple of q. Therefore, c = p*(Some integer) ........ (I)
Question: Is c divisible by a/d?
a/d = d*p/d = p
Question: Is c divisible by p?
From (I) above, we know that c is a multiple of p and hence is divisible by p. So 'YES, c is divisible by a/d.'
Answer (C)