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GRE Math: Study Guide & Test Prep27 chapters | 182 lessons | 16 flashcard sets

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Lesson Transcript

Instructor:
*Laura Pennington*

Laura has taught collegiate mathematics and holds a master's degree in pure mathematics.

In this lesson, we'll define the division algorithm and divisibility. We'll see how these two concepts are related and use examples to explore some different divisibility rules to add to your math toolbox.

Remember in elementary school when you would bring a treat in to share with the class on your birthday? Wasn't that great? Suppose it's your birthday, and you decide to keep tradition alive and bring in 25 pieces of candy to share with your coworkers. You have 6 coworkers in your department to whom to give the candy. You sit down to figure out how many pieces of candy each worker will receive.

You realize this is a simple division problem. You divide the number of pieces of candy by the number of coworkers to solve the problem.

25 / 6 = 4 remainder 1

This tells you that each coworker will get 4 pieces of candy, and you will have 1 piece leftover. In other words:

25 = 6 * 4 + 1

This equation actually represents something called the **division algorithm**. In the equation, we call 25 the **dividend**, 6 the **divisor**, 4 the **quotient**, and 1 the **remainder**. The division algorithm is basically just a fancy name for organizing a division problem in a nice equation. It states that for any integer *a* and any positive integer *b*, there exists unique integers *q* and *r* such that *a* = *bq* + r, where *r* is greater than or equal to 0 and less than *b*. Does that equation look familiar? It should! It's exactly in the form of the equation we found representing our candy problem! Now, let's talk about a special case of the division algorithm: that is, when we have a remainder equal to 0.

When we set up a division problem in an equation using our division algorithm, and *r* = 0, we have the following equation:

*a* = *bq*

When this is the case, we say that *a* is divisible by *b*. If this is a little too much technical jargon for you, don't worry! It's actually fairly simple. If a number *b* divides into a number *a* evenly, then we say that *a* is divisible by *b*.

For example, 8 is divisible by 2, because 8 / 2 = 4. However, 8 is not divisible by 3, because 8 / 3 = 2 with a remainder of 2. We see that we can check to see if a number, *a*, is divisible by another number, *b*, by simply performing the division and checking to see if *b* divides into *a* evenly.

Pretty cool, huh? Well, it's about to get even cooler! Let's look at some really neat and fun rules that we can use to determine if certain numbers will divide into other numbers evenly.

There are some pretty brilliant divisibility rules that will tell us about specific numbers and their divisibility. For instance, you may realize that even numbers are always divisible by 2. This is just one of many divisibility rules. Here are some more of the simpler ones:

- If you have the
**number 3 as the divisor**, the dividend is divisible by divisor if the sum of the digits is divisible by 3. For example, the number 27 is divisible by 3 because the sum of the digits, 2 + 7 = 9, which is divisible by 3, so 27 is divisible by 3.

- If you have the
**number 4 as the divisor**, the dividend is divisible by divisor if the last two digits of the dividend are divisible by 4. For example, the number 328 is divisible by 4 because its last two digits, 28, are divisible by 4.

- If you have the
**number 5 as the divisor**, the dividend is divisible by divisor if the last digit is 0 or 5.For example, the number 30 is divisible by 5 because the last digit in 30 is 0. - If you have the
**number 6 as the divisor**, the dividend is divisible by divisor if the number is divisible by both 2 and 3. For example, the number 24 is divisible by 6 because 24 is divisible by both 2 and 3. - If you have the
**number 10 as the divisor**, the dividend is divisible by divisor if the last digit is 0. For example, the number 270 is divisible by 10 because it ends in 0.

There are many more of these rules for different numbers, but these are some of the more common and simpler ones. Let's take a look at an example pulling all this together.

Let's revisit the candy at work example. Suppose that you are trying to decide what package of candy to buy to bring to work to pass out to your 6 coworkers. One package has 36 pieces of candy in it, and the other one has 44. You don't want to have any pieces left over. Any ideas as to how this relates to the division algorithm and divisibility?

Well, we know we can determine how many pieces of candy each worker will get by performing division, and we don't want any pieces leftover. In the division algorithm, this means we want the remainder to be 0. In turn, this tells us that we want the number of pieces of candy to be divisible by 6! Ah-ha! That's the connection! So, really, all we have to do to decide which candy to buy is determine if 36 and 44 are divisible by 6.

We can perform the division, or we can use the divisibility rule for 6, which states that the dividend must be divisible by both 2 and 3. The number must be even to be divisible by 2, and the sum of the digits must be divisible by 3 to be divisible by 3. We see that both 36 and 44 are even, so they are both divisible by 2. Easy enough! Let's look at the sum of their digits.

To see if 36 is divisible by 6, we add the two digits together and then see if that sum is divisible by 3. 3 + 6 = 9, and 9 / 3 = 3, so 36 is divisible by both 2 and 3. Therefore, 36 is divisible by 6.

Now, let's check to see if 44 is divisible by 6. We'll add the two digits together and then see if that sum is divisible by 3. 4 + 4 = 8, and 8 / 3 = 2 with remainder of 2, so 44 is divisible by 2 but not by 3. Therefore, 44 is not divisible by 6.

We see the sum of the digits of 36 is divisible by 3, but the sum of the digits of 44 is not divisible by 3. Therefore, 36 is divisible by 6 and 44 is not. Well, that made the candy decision much easier!

The **division algorithm** states that for any integer, *a*, and any positive integer, *b*, there exists unique integers *q* and *r* such that *a* = *bq* + *r* (where *r* is greater than or equal to 0 and less than *b*). We call *a* the **dividend**, *b* the **divisor**, *q* the **quotient**, and *r* the **remainder**. When the remainder is 0, we say that *a* is divisible by *b*. A number, *a*, is divisible by a number, *b*, when *b* divides into *a* evenly.

Being familiar with divisibility and the division algorithm helps us to understand division even more than we already do! Therefore, these concepts are great to have in your math toolbox.

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GRE Math: Study Guide & Test Prep27 chapters | 182 lessons | 16 flashcard sets

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