Divisibility

[JackH825]

Answer:
Source: GMAT official Qs (Quantitative :: Problem solving:: 03344)

How many positive three-digit integers are divisible by both 3 and 4 ?
75
128
150
225
300

Please can you let me know the most efficient way to answer Qs such as the above.

Thanks!

[Sage Pearce-Higgins]

I encourage you to "think in prime factors". So, divisible by 3 and 4 means that the number has prime factors of 2,2,3 (and possibly others, of course). Any number with these prime factors is divisible by 12. So the question can be rephrased as 'how many numbers from 100 through 999 are divisible by 12?'

Since the answers are spread out, we can afford to be pretty sloppy with our calculation. Something like: there are 8 multiples of 12 up to 100, so let's say that there are 8 multiples of 12 in every 100 numbers. That would make 8x9 between 100 and 999 = 72, close enough to give me answer A.

For a more precise method, simply divide 999 by 12 and discard the remainder to find the number of multiples of 12 up to that number = 83. Now subtract the multiples of 12 up to 100, that's 8 of them to give you 75.