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What does the GMAT mean when it says "divisible"?

Problem Solving — algebra and arithmetic (GMAT Focus Edition)
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by stevennu » Sun Sep 01, 2013 10:37 pm
If x is a positive integer, is (x)(x + 2)(x + 4) divisible by 12?

(1) x^2 + 2x is a multiple of 3.

(2) 3x is a multiple of 2.

I answered D when in fact the answer is B. My reasoning was as follows

(1) If X^2 + 2X is a multiple of three, then I can assume that the equation can look something like the following: 3(x+4) or 6(x+4) or 9(x+4). In each of these cases, the result is a number that is divisible by 12. Not necessarily evenly divisible, but definitely divisible. Therefore A is sufficient.

On to (2) 3X is a multiple of 2.

X could equal, 2, 4, 6, 8, etc. Plugging these numbers into the original equation yields numbers that are all evenly divisible by 12.

When the GMAT specifies divisible by does it mean evenly without remainders?

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by fourteenstix » Mon Sep 02, 2013 8:13 am
First, in this particular question, the GMAT wants you to recognize that(x)(x + 2)(x + 4) is the same thing as saying "3 consecutive even integers" or "3 consecutive odd integers". Whether x is even or odd will determine which is true. If x is even, these are three consecutive even integers (e.g. x=2, x+2=4, x+4=6). Conversely, if x is odd, these are three consecutive odd integers (e.g. x=1, x+2=3, x+4=5). It's worth remembering that three consecutive even integers will be divisible by 2, 3, and 4. It will be divisible by 2 because the product will always be even (even * even * even = even), it will be divisible by 3 because there are three numbers (the product of n numbers will always be divisible by n), and it will be divisible by 4 because one of the three terms must be divisible by 4 (you cannot find 3 consecutive even integers in which one of the integers is not divisible by 4).

To say that the product of these three numbers is divisible by twelve is to say: When these three numbers are multiplied together, can they evenly be divided by 12? Or, is the product of these three integers a multiple of 12 (12, 24, 36, etc.). Or, if you took the product of these three numbers, and then divided it by 12, would the remainder be 0 (as you pointed out at the bottom of your post)?

Statement one tells us that the product of the first two numbers (x and x+2) is a multiple of three; however, we are not told whether x is even or odd. Without knowing whether x is even or odd, you'll see that we cannot answer whether this product is divisible by 12 with certainty in your examples above:

X=ODD
3(3+4)=21 Divisible by 12? NO
X=EVEN
6(4+4)=48 Divisible by 12? YES

Already we see that we have insufficiency to satisfy the stem.

Statement 2: If 3x is a multiple of two, we know that x MUST be even:

3 * X = 2, which means
Odd * X = Even, therefore
X must be even

While the question is literally asking if the product of these numbers is divisble by 12, we see now that it's really as simple as determining whether x is even or not. That's precisely what Statement 1 failed to do and what Statement 2 has done.

I hope this helps!
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by stevennu » Mon Sep 02, 2013 10:07 am
So is it a hard and fast rule that when the GMAT states "divisible by" it means divisible with remainder 0?
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by Brent@GMATPrepNow » Mon Sep 02, 2013 10:15 am
stevennu wrote:So is it a hard and fast rule that when the GMAT states "divisible by" it means divisible with remainder 0?
Exactly.

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Brent
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by fourteenstix » Mon Sep 02, 2013 10:50 am
stevennu wrote:So is it a hard and fast rule that when the GMAT states "divisible by" it means divisible with remainder 0?
Yep. It will always be an integer (remainder of 0).
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by Brent@GMATPrepNow » Mon Sep 02, 2013 10:58 am
stevennu wrote:So is it a hard and fast rule that when the GMAT states "divisible by" it means divisible with remainder 0?
I should point out that there are many different ways to express the concept of divisibility.

For example, saying that "integer x is divisible by integer y" is the same saying:
- "when x is divided by y the remainder is zero.
- "y is a divisor/factor of x"
- "x = ky for some integer k"
- "x is a multiple of y"
- "y is hiding in the prime factorization of x"

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