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Answer given is: One of any three consecutive even integers is divisible by 3. Because this integer is even, it is also divisible by 6. When multiplied by two more even integers, it renders a product that is divisible by 24. S_1 is sufficient. S_2 is not sufficient. We need to know something about c.
The correct answer is A.
My question What about 0? i.e. 0, 2, 4, shouldn't the answer be E?
I've come across a question that I'm a bit confused by and would appreciate your help with.
Question: Is a*b*c divisible by 24 ?
1. a, b, and c are consecutive even integers 2. a*b is divisible by 12
GMATClub Answer: One of any three consecutive even integers is divisible by 3. Because this integer is even, it is also divisible by 6. When multiplied by two more even integers, it renders a product that is divisible by 24. S_1 is sufficient.
S_2 is not sufficient. We need to know something about c.
The correct answer is A.
-----
The GMATClub answer seems to be assuming that the consecutive integers are positive. Is this not an incorrect assumption? The question doesn't state the even integers are positive.
As an example: the values for a, b and c could be 2, 4 and 0, whose product is not divisible by 24. Or the values could be 2, 4 and 6, for which the product is divisible by 24. So S1 seems insufficient.
Could someone let me know if I've misunderstood this? Many thanks in advance,
Re: M13 Q 30 - Data Sufficiency & Number Properties [#permalink]
23 Feb 2009, 07:33
I think 0 is divisible by any number... not sure though and -ve numbers I dont think make a difference here as only the sign will change but the number will remain divisible by 24.
Another way of phrasing the question would be: is abc/24 an integer?
Recall that an integer is a whole number in the infinite set {. . . -3, -2, -1, 0, 1, 2, 3, . . .}.
Statement 2: If abc is divisible by 12, then it could also be divisible by 24 if the the number equalled 24, 48, 240, etc. But if abc is 12, then the number is not divisible by 12. In short, we can't determine the answer. NOT SUFFICIENT.
Statement 1: Let's start with the smallest absolute values allowed first: 0, 2, and 4. When multiplied, this equals zero - an integer. When zero is divided by any non-zero integer, the result is zero - again, an integer. So it's divisible by 24. If we use, negative numbers -2, -4, -6, we arrive at -48/24 = -2, an integer. If we break down 24 into primes, we have one 3 and three 2's. Pick any consecutive set of even integers. These prime numbers will be in there. In other words the number is divisible by 24. SUFFICIENT.
And the GMAT assumes that you know about the divisibility rules of 0, just as it assumes that you know about the rules for 3, 2, 10, 5, etc. _________________
Answer given is: One of any three consecutive even integers is divisible by 3. Because this integer is even, it is also divisible by 6. When multiplied by two more even integers, it renders a product that is divisible by 24. S_1 is sufficient. S_2 is not sufficient. We need to know something about c.
The correct answer is A.
My question What about 0? i.e. 0, 2, 4, shouldn't the answer be E?
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 yes, 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: th 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 given is: One of any three consecutive even integers is divisible by 3. Because this integer is even, it is also divisible by 6. When multiplied by two more even integers, it renders a product that is divisible by 24. S_1 is sufficient. S_2 is not sufficient. We need to know something about c.
The correct answer is A.
My question What about 0? i.e. 0, 2, 4, shouldn't the answer be E?
This is how I went about solving the problem... Statement 1) I factored 24 and got 2, 2, 2, 3 I used a MGMAT strategy that an even x even = divisible by 4 even x even x even = divisible by 8 Since there are three consecutive even numbers I knew it was even x even x even (at least divisible by 2,2,2) and since there are three terms it has to be divisible by 3
Statement 2) is insufficient because there is no mention of C. If a*b = 12 and c is 1 then the answer yields NO. However, if we make a*b = 24 and c = 1 we get an answer of YES. Thus, insufficient. _________________
I'm trying to not just answer the problem but to explain how I came up with my answer. If I am incorrect or you have a better method please PM me your thoughts. Thanks!
Clearly, Statement 1 is Sufficient.Any three consecutive evern numbers will have 3 multiple of 2 that comes to 2*2*2 = 8.Also, any three consecutive evern numbers will have atleast 1 multiple of 3, so that leads to 8 (from above)*3=24.Hence, any three consecutive even no is divisible by 24. If we take 0 as one of the even nos among the three, then also 0 is divisible by any no >0.Hence Statement 1 Is sufficient.
Statement 2 :- Insufficient, Imagine 3,4 and 1 as one of the sets that is divisible by12 but not by 24.Hence, Insufficient. I would rate this question as 600 level and nothing beyond for sure.
Another way of phrasing the question would be: is abc/24 an integer?
Recall that an integer is a whole number in the infinite set {. . . -3, -2, -1, 0, 1, 2, 3, . . .}.
Statement 2: If abc is divisible by 12, then it could also be divisible by 24 if the the number equalled 24, 48, 240, etc. But if abc is 12, then the number is not divisible by 12. In short, we can't determine the answer. NOT SUFFICIENT.
Statement 1: Let's start with the smallest absolute values allowed first: 0, 2, and 4. When multiplied, this equals zero - an integer. When zero is divided by any non-zero integer, the result is zero - again, an integer. So it's divisible by 24. If we use, negative numbers -2, -4, -6, we arrive at -48/24 = -2, an integer. If we break down 24 into primes, we have one 3 and three 2's. Pick any consecutive set of even integers. These prime numbers will be in there. In other words the number is divisible by 24. SUFFICIENT.
And the GMAT assumes that you know about the divisibility rules of 0, just as it assumes that you know about the rules for 3, 2, 10, 5, etc.
Regarding Statement 2: since a*b is divisible by 12...e can write a*b as 12*X(considering X as integer)------(1). since c is an even number c can be written as 2*Y(considering Y as integer)----------(2).
From (1) and (2) e can write a*b*c = 12*X*2*y=24XY .......so a*b*c is equal to 24XY hich is obviously divisible by 12
so i think statement 2 alone is also SUFFICENT.....Please correct me if iam wrong