dexerash wrote:

Is a*b*c divisible by 24?

(1) a, b, and c are consecutive even integers

(2) a*b is divisible by 12

This problem is from GMATClub Quant tests.

I am confused in this problem. Since 0 is also an EVEN number so statement 1 cannot be true. Am i right?

Please help!

Note that an integer \(a\) is a multiple of an integer \(b\) (integer \(a\) is a divisible by an integer \(b\)) means that \(\frac{a}{b}=integer\): so, as 0 divided by any integer (except zero itself) yields an integer then

zero is a multiple of every integer (except zero itself).

Also on GMAT when we are told that \(a\) is divisible by \(b\) (or which is the same: "\(a\) is multiple of \(b\)", or "\(b\) is a factor of \(a\)"), we can say that:1. \(a\) is an integer;

2. \(b\) is an integer;

3. \(\frac{a}{b}=integer\).

BACK TO THE ORIGINAL QUESTION:

Is a*b*c divisible by 24?(1) a, b, and c are consecutive

even integers --> \(a=2k-2\), \(b=2k\) and \(c=2k+2\) for some integer \(k\) --> \(abc=(2k-2)2k(2k+2)=8(k-1)k(k+1)\), now \((k-1)\), \(k\), \((k+1)\) are 3 consecutive integers, which means that one of them must be a multiple of 3, thus \(abc\) is divisible by both 8 and 3, so by 24. Sufficient.

Or even without the formulas: the product of 3 consecutive even integers will have 2*2*2=8 as a factor, plus out of 3 consecutive even integers one must be a multiple of 3, thus abc is divisible by both 8 and 3, so by 24.

(2) a*b is divisible by 12, clearly insufficient as no info about c (if ab=12 and c=1 answer will be NO but if ab=24 and c=any integer then the answer will be YES).

Answer: A.

Hope it's clear.

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