# Basic GRE Math Concepts: Multiples and Divisibility

Finding the factors of a number will help you tremendously on test day because you will very often be reducing fractions. Lets see how this works in an example.

You must know how to reduce fractions on the GRE, and finding a common factor is a good way to reduce fractions. Lets say we have the fraction 24/36.  If we list the factors of 24 and 36 we have a menu to choose from to find a common factor. It looks like the Greatest Common Factor is 12. So, in order to reduce the fraction, we divide the top by 12 and the bottom by 12. Thus, if we divide 24 by 12 we get 2 and if we divide 36 by 12 we get 3. So 24/36 reduces to 2/3.

Now, if you were to try to reduce 24/36 in your head, you might think of 6 as a common factor and try to reduce it using that. 24/6 is 4 and 36/6 is 6 thus we have 4/6 left. But, this fraction can be reduced even more because both the top and bottom number are even, we can divide them by two. Thus 4/6 equals 2/3.

The point is that you can reduce a fraction as long as there is a common factor. It doesn’t always have to be the Greatest Common Factor. And keep reducing until there is no common factor for the numerator and the denominator.

Again, finding a common factor of two numbers is a crucial skill you need to know for test day. Many GRE questions can be answered by simply reducing the fraction and not by doing complicated division.

As an illustration, lets examine this quantitative comparison question: which column contains the bigger quantity?

Here we have 24/36 versus 22/33. Most people would not be able to compare these directly in their head. Since the denominators are different, we would have some difficulty comparing them directly. Instead, if we reduce the fractions, we may be able to tell which is bigger.

For column A, we discovered on the previous slide that 24/36 reduces to 2/3… What about column b? What is a common factor of the 22 and 33? 11 would work. And it may be the greatest common factor. So, if we divide the top by 11 and the bottom by 11, we get 2/3. Lo and behold, we get the same value for both columns. The columns are equal so our answer choice is C.

Key point: make sure you are comfortable reducing fractions on test day.

Having covered factors and multiples, we can now move on to a related topic, divisibility. This is an extremely important section because often times on the GRE, you will find questions which ask directly about divisibility, so make sure you pay careful attention to what we are going to go over here.

First, lets talk about what divisibility is. The number 10 is divisible by 5 because 5 goes into 10 two times nicely.10/5 = 2.

12 divided by 5, on the other hand, does not divide so nicely. Thus when you divide 12 by 5, you get 2 with a remainder of 2. If you wanted to write that remainder in fractional form, simply put the remainder over what you divided by in the first place (in this case the 5). So our fractional form of this answer is 2 and 2 fifths, 2 2/5.

The upshot of this is that 12 is not divisible by 5 because it does not divide a whole number of times. 10, however, is divisible by 5, because it divides a whole number of times.

Let’s see if we can find a pattern to determine divisibility… For example, I know 14 is divisible by 7. It divides two times. Thus 15 divided by 7 is 2 with 1 remainder, or 1/7… And 16 divided by 7 is 2 with a remainder of 2, or two sevenths… The next number divisible by 7 is 21  with no remainder….

What is the pattern to see if a number divides evenly into another number?

A larger number is divisible by a smaller number if and only if the larger number is a multiple of the smaller number. We know 21 is divisible by 7 because 21 is 7 times 3.

Let’s talk a little bit more about multiples…

A multiple is just an integer times a counting number. For example, the multiples of 3 are 3×1, 3×2, 3×32, 3×4, 3×5 etc.

These work out to 3, 6, 9, 12, 15, and so on. You can think of the multiples of 3 as simply counting by 3s.

The multiples of 5, then, are 5, 10, 15, 20, 25, .. And so on.

The multiples of 7 are 7, 14, 21, 28, 35, and so on.

This is important because of the kinds of questions you might see on the GRE.

For example, sometimes you will need to find a COMMON MULTIPLE between two numbers, often times the LEAST common multiple. (This often necessary when adding or subtracting fractions, which will be discussed in a later). In this instance, we are asked what is the least common multiple of 5 and 7? In other words, if we were to count by 5s and count by 7s, where do the numbers converge? Is there a number that is common to the multiples of 5 and the multiples of 7?

Well, if we look back at our lists, we see the least common multiple of 5 and 7 is 35.

Keep in mind that there are an infinite number of common multiples between two integers. For example, 70 is another common multiple, as is 105, 140, and so on.That is why it is customary to look for the least common multiple.

You can always find a common multiple by multiplying the 2 numbers together, in this case 5 times 7. It is not a guarantee that it will always be the LEAST common multiple between the two numbers, however. Let’s try another example.

What is the least common multiple between 5 and 10?

Turns out, 10 is the least common multiple. Every number is a multiple of itself. In this case 10 is 10×1.

Multiples are a very simple concept, just think of a number times an integer. Again, every number is a multiple of itself.

So, it turns out that multiples and factors are basically two sides of the same coin. If you will remember from a few slides back, two of the factors of 12 are 6 and 2. What that means is 12 is a multiple of 6, as well as a multiple of 2. That is, 12 is evenly divisible by 6 and 2. It divides nice and neat with no remainder.

Knowing multiples and divisibility will enable you to reduce fractions, which is a key component for doing well on the GRE math.

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