Thoughts of standing at a chalkboard and completing what seems like endless long division problems may have your students scared to tackle division. While thinking in terms of multiplication and division is different than addition and subtraction, your students can grasp it. For many, having a clear set of rules and structure helps to clarify the concept and give students a hand when they try to solve equations. As you begin your section on division, be sure to share these rules with your class and discuss them as a part of your math talk:

### Divisibility by 2

If the last digit in a number is 0 or an even number, it’s divisible by 2. For instance, 20 ends in a 0. When divided in half, the result is 10, which is an even number. When students see a number that ends in an even digit, they should know the number can be divided into two evenly. However, numbers that end in odd digits can be divided by two, they’ll just have a remainder or decimal.

**Divisibility by 3**

A number is divisible by 3 if the sum of the digits is divisible by 3. To use this trick, students must have some ability to divide, but checking smaller numbers is less daunting than a large one. For instance, if you ask students if 168 is divisible by 3, they should do the following:

1 + 6 + 8 = 15

15/3 = 5

Therefore, 168 is divisible by 3.

### Divisibility by 4

If the last two digits of a number are divisible by 4, the whole number is. For example, in 1,012, 12 is divisible by 4. However, in 1,013, 13 is not.

**Divisibility by 5**

When the last digit of a number is 0 or 5, the number can evenly be divided by 5. As such, 5, 10, 15, 20, 25 and so on can all be divided by 5. Students can look at large numbers and say right away whether it can evenly be divided into five parts.

### Divisibility by 6

Numbers divisible by 6 can also be divided by both 3 and 2. Students should test the number with both rules for 3 and 2. If the number passes both tests, it can be divided by 6. If it fails just one, test it cannot. For instance:

308 ends in an even digit, so it’s divisible by 2. However, 3 + 0 + 8 = 11, which cannot be divided evenly by 3. As such, 308 is not divisible by 6.

**Divisibility by 8**

“The divisibility rule for 9 is the same as for 3.”

A large number is divisible by 8 if the last three digits are also divisible by 8 or are 000. In 7,120, 120 can be divided evenly by 8, so the whole number is divisible by 8 as well.

### Divisibility by 9

The divisibility rule for 9 is the same as for 3, which makes sense given that 9 can be divided by 3. If the sum of a number’s digits is divisible by 9, so too is the entire number. For example:

In 549, 5 + 4 + 9 = 18

18/9 = 2

So, 549 is divisible by 9.

### Divisibility by 10

If the last digit is 0, the number can be divided evenly by 10.

**Why the rules help and how to use them**

These rules allow students to look at larger numbers in a less-daunting context. Divisibility rules also let them learn a lot about a number by simply looking at its digits. As such, you should encourage students to use all rules when examining a number. When looking at something like 1,159,350, students can go down the divisibility list, checking off which numbers the large one can be divided by.

Of course, you won’t only talk about even division in your math class. Some numbers will have remainders. You can still use the rules to talk about those numbers. Have students determine whether a certain number will have a remainder when divided by 2, 3, 4, 5, 6, 8 or 10.

For lesson plan ideas that utilize divisibility rules, take a look at our book “Math Matters: Understanding the Math You Teach, Grades K–8, Second Edition,” and visit our website to find additional, free resources.

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