Data Sufficiency

If m and n are positive integers, then is 5m+2n divisible by 3m+n?

(1) m is divisible by n
(2) m is divisible by 15 and n is divisible by 2

OA is B

Source: NOVA's GMAT DS Prep Course, p.135

The explanation in the book is very long. I tried to solve this one by testing numbers, but it took me more than 3-4 minutes. Is there any shortcut to solve this one?

B) m=15x and n=2y
75x+4y / 45x+2y.
75x+4y = p(45x+2y)
(75-45p)x = (2p-4)y
x = (2p-4)/(75-45p) *y
m,n positive. thus (2p-4)/(75-45p) also positive.
either (2p-4)<0 &(75-45p)<0 or (2p-4)>0 & (75-45p)>0
either: p<2. but p is positive integer. thus p=1.75-45=30<0 false. thus p is not equal to 1.
or: p>2. but then (75-45p)>0 is false. Thus no solution.
Thus not divisible.
Sufficient

A) m=nx.
5x+2 / 3x+1
5x+2 = p(3x+1)
x = p-2 / 5-3p
x>0
either: p>2 and 5-3p>0 not possible.
or p<2 5-3p<0 not possible.
IMO D