What is the greatest positive four-digit integer which must be added

Prime factorising the divisors:

12: \(3*4\)
15: \(3*5\)
27: \(3^3\)
32: \(2^5\)
40: \(5*2^3\)

LCM: \(2^5 * 3^3 * 5\)

\(32 * 27 * 5 = 4320\) The answer will be the choice which when added to \(5231\) yields a multiple of \(4320\).

Looking at the sum of \(5231\) and each of the answer choices, their respective sums will range somewhere between \(11 000 \) and \(16 000\). Multiples of \(4320\) which are 5 digits big will \(4320*3\) and \(4320*4\). However, \(4320*4\) will take us our of the range of between \(11 000 \) and \(16 000\).

\(4320*3 = 12960\) will be the answer.

Glancing at the answer choices: Adding \(5231\) to (A), (B) & (C) will send us over \(12960\). Adding \(5231\) to (E) will put us under. (D) is the safest place to start.

(D) \(5231+7729 = 12 960\)

Answer D

We could also notice that, using any answer choice, we'll get a multiple of 5 and 4, because the choices all have the same last two digits. So we probably don't need to worry about 2s and 5s here. But we need also to get a multiple of 3 when we add the right answer choice, and that's not going to happen with most of the choices. The number 5321 gives a remainder of 2 when you divide by 3 (since the sum of its digits is 11), so we need to add something with a remainder of 1 when divided by 3 if we want the sum to be divisible by 3. Summing the digits of each answer choice, only D, 7729, gives a remainder of 1 when divided by 3, so it must be right.