If 5n + 4 is divisible by 3 and 6m + 2 is divisible by 5 (m and n are

Bunuel is there a more scientific way to solve this? Would very much appreciate to better get the logic behind this problem. Thanks a lot!

If 6m + 2 is divisible by 5, then so is (6m + 2) - 5m, because if you subtract one multiple of 5 from another, you always get a multiple of 5. So m + 2 is a multiple of 5, and the units digit of m must be 3 or 8.

I wouldn't try to be any more scientific than that for this particular question, because it now only takes a few seconds to work out the right answer -- there are only two answer choices left, and if we test one and it's wrong, the other is right.

Conceptually it's really a question about remainders, though I won't go into the theory in too much detail because it's not very helpful for this problem. But notice above, if the units digit of m is 3 or 8, the remainder is 3 when we divide m by 5. The information about 5n + 4 will similarly give us the remainder when we divide n by 3 (it means, if you do a bit of work, that the remainder is 1 when you divide n by 3). Since m and n are the same number, the question is really asking "if the remainder is 3 when you divide m by 5, and the remainder is 1 when you divide m by 3, what is the smallest positive value of m?" which is a variation on a standard remainder question format on the GMAT (for an official example, you can see this problem from the OG). My Number Theory book explains what I think is the easiest way to solve problems like this, but there are a few different methods.

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