I'm assuming the problem is supposed to say "largest" rather than "large." Anyway, nearly every time a problem talks about factors, or divisibility, a good first thought is to factor out the number down to its primes. I would first see whether the number meant anything to me. For example, if in this problem, quantity A said "343", I would recognize that as 7 cubed and thus the answer would be C. This is why it's a good idea to study the squares, cubes, and powers.
In this problem though, the number 1,728 doesn't mean anything to me, so I'll have to break it down to its primes. Now, 1,728 may seem pretty large but it'll go pretty quickly. I'd divide by 2 every time first because it's the easiest to divide by:
1,728 > 2x864 > 2x2x432 > 2x2x2x216 > 2x2x2x2x108 (at this point I recognize that 108 is 27x4, and the largest prime factor of 27 is 3, but if you don't recognize 108, just keep going until you recognize the number) > 2x2x2x2x2x54 > 2x2x2x2x2x2x27 > a bunch of 2s and three 3s.
Therefore, the largest prime of 1,728 is 3 and the answer is B.
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