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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