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## Zero is divisible by

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The concept of "multiples" can be confusing. If x is a multiple of y, that means x is *evenly divisible* by y. 16 is a multiple of 2, 100 is a multiple of 50, and 72 is a multiple of 72.

In number theory, it is conventional that "factors" and "multiples" refer to positive numbers. (Depending on the context, this varies, but it will suffice for the purposes of the GMAT.) While -20 is evenly divisible by 4, -20 is not considered a "multiple" of 4.

By the same reasoning, zero is generally not considered a multiple of anything.

The *concept* of multiples arises frequently on the GMAT, but the exam doesn't often use the word. Sometimes multiples are implied in a word problem. On other items, the GMAT uses the more direct language "divisible by."

In many ways, "x is a multiple of y" and "x is divisible by y" are equivalent. However, the latter allows for more possible values of x. Negative numbers are divisible by some positives (as in the example above of -20, evenly divisible by 4), and zero is divisible by everything.

Let me emphasize that point. **Zero is divisible by everything.**

Another way of thinking about divisibility is this: x is evenly divisible by y if, when x is divided by y, there is no remainder. When 35 is divided by 7, there is no remainder, so 35 is evenly divisible by 7.

When dividing zero by any number, the result is zero, with no remainder. Thus, by the definition of divisibility, zero is divisible by everything.

On all questions, but especially Data Sufficiency, the GMAT loves testing to see whether you are aware of these "special cases," which often include zero. When considering divisibility, remember the one integer that isn't either positive or negative.

**About the author:** *Jeff Sackmann has written many
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Total GMAT Verbal, and GMAT 111. He has also created
explanations for problems in The Official Guide, as well as
1,800 practice GMAT math questions.*

Total GMAT Math
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