Fewer Factors, More Multiples

Sometimes it is easy to confuse factors and multiples. The mnemonic “Fewer Factors, More Multiples” should help you remember the difference. Factors divide into an integer and are therefore less than or equal to that integer. Positive multiples, on the other hand, are 1 times that integer, 2 times that integer, etc. and are therefore greater than or equal to that integer.

Any integer only has a limited number of factors. For example, there are only four factors of 8: 1, 2, 4, and 8. By contrast, there is an infinite number of multiples of an integer. For example, the first five positive multiples of 8 are 8, 16, 24, 32, and 40, but you could go on listing multiples of 8 forever.

Factors, multiples, and divisibility are very closely related concepts. For example, 3 is a factor of 12. This is the same as saying that 12 is a multiple of 3, or that 12 is divisible by 3.

On the GRE, this terminology is often used interchangeably in order to make the problem seem harder than it actually is. Be aware of the different ways that the GRE can phrase information about divisibility. Moreover, try to convert all such statements to the same terminology. For example, all of the following statements say exactly the same thing:

• 12 is divisible by 3
• 12 is a multiple of 3
• is an integer
• 12 = 3n, where n is an integer
• 12 items can be shared among 3 people so that each person has the same number of items
• 3 is a divisor of 12, or 3 is a factor of 12
• 3 divides 12
• yields a remainder of 0
• 3 “goes into” 12 evenly

Another term that the GRE sometimes uses is “unique prime factor.” The distinction between a prime factor and a unique prime factor is best illustrated by an example. If you prime factor 12, you end up with two 2’s and one 3, but 12 only has two unique prime factors, 2 and 3, because the two 2’s are the same number. So 100 has two unique prime factors (2 and 5) just as 10 does.