Can I divide this number by

that number(without leaving a remainder)?

There’s an easy way to tell! Use divisibility rules.

DIVISIBILITY means

one whole number (…-3, -2, -1, 0, +1,+2, +3…)

can be divided by another whole number

without leaving a remainder or

(to put it another way)

leaving a remainder of 0.

Divisibility Rules

A number can be divided by…. If…

1 all of the time.

2 its last digit is even: 0, 2, 4, 6, or 8.

3 the sum of its digits is divisible by 3.

4 its last two digits (viewed as a two-digit number) are divisible by 4.

5 it ends in 0 or 5.

6 it is even and the sum of its digits is divisible by 3.

7 the original number without the last digit minus 2x last digit, repeated until the difference is 20 or less provides an answer (difference) divisible by 7.

8 its last 3 digits (taken together as a three-digit number) are divisible by 8.

9 the sum of its digits is divisible by 9.

10 it ends in 0.

A number divided by 0 is always undefined.

Can I divide this number by 1?

YES!EVERY NUMBER can be divided by 1.

In fact, the number 1 is quite magical, because when you divide any number by 1, the quotient (answer) is ALWAYS the number you started with.

328 ÷ 1 = 328This works in multiplication too.

328 x 1 = 328And this special number magic has a name. The

Multiplicative Identity Property. (We’ll learn more about that later!)

Can I divide this number by 2?

YES

If the last digit in the number is even.

Look at the last digit in the number.

If it is a 0, 2, 4, 6, or 8

the number is even and you can divide it by 2!

No

If the last digit in the number is odd.

Look at the last digit in the number.

If it is a 1, 3, 5, 7, or 9

The number is odd and you can NOT divide it by 2!

Is 328 divisible by 2?

What about 328?

Use the rule.• The last digit in 328

is 8.• 8 is an even

number.

SO, I know that

328 is divisible by 2.

328 ÷ 2 = 164

Is 327 divisible by 2?

Use the rule.

1. The last digit in 327 is 7.

2. 7 is an odd number.

SO, I know that

327 is NOT divisible by 2.

327 ÷ 2 = 163 R1

Can I divide this number by 3?YES

If the sum of the digits in the number is divisible by 3.

Is 327 divisible by 3?

Use the rule.

1. Add the digits of 327 together.

2. 3 + 2 + 7 = 12

3. Divide the sum of the digits by 3.

4. 12 ÷ 3 = 4 Remainder 0

5. There is no remainder.

SO, I know that

327 IS divisible by 3!

See. It is.

109 R 0

3) 327 3 + 2 + 7 = 12

- 3 12 ÷ 3 = 4 R 0

02

- 0

27

-27

0

Is 329 divisible by 3?

Use the rule.

1. Add the digits of 329 together.

2. 3 + 2 + 9 = 14

3. Divide the sum of the digits by 3.

4. 14 ÷ 3 = 4 Remainder 2

5. There is a remainder…

SO, I know that

329 IS NOT divisible by 3!

See. It isn’t.

109 R 2

3) 329 3 + 2 + 7 = 12

- 3 12 ÷ 3 = 4 R 0

02

- 0

29

-27

2

Can I divide this number by 4?YES

If the number formed by the last two digits is divisible by 4.

(YIKES! What does that mean?)

Ignore all the digits in the number, but the last two.

Look at the last two digits as a number.

Can you divide it by 4?Good. Then you can divide the

entire number by 4.

Is 327 or 328 divisible by 4?

Use the rule.1. The last two digits

in 328 are 28.2. 28 is divisible by 4.28 ÷ 4 =7 Remainder 0SO, I know that 328 is divisible by 4.

328 ÷ 4 = 82

Use the rule.1. The last two digits

in 327 are 27.2. 27 is NOT divisible

by 4.27 ÷ 4 = 6 Remainder 3SO, I know that327 is NOT divisible by

4.327 ÷ 4 = 81 R3

Can I divide this number by 5?

YES

If the digit in the ones place (the last digit to the right) is a 0 or a 5.

Is 325 or 330 or 328 divisible by 5?Use the rule.

1. The last digit in 325 is 5.

So, I know that

325 is divisible by 5.

325 ÷ 5 = 65

and

Use the rule.

1. The last digit in 330 is 0.

So, I know that

330 is divisible by 5.

330 ÷ 5 = 66

Use the rule.

1. The last digit in 328 is not 0.

2. The last digit in 328 is not 5.

3. The last digit in 328 is 8.

So, I know that

328 is NOT divisible by 5.

328 ÷ 5 = 65 R3

Can I divide this number by 6?

YES

If the number is divisible by both 2 and 3.

NO

If the number is divisible by 2, but NOT by 3.

If the number is divisible by 3, but NOT by 2.

Is 328 divisible by 6?Is 328 divisible by 2?Use the rule.1. The digit in the ones

place is 8. 2. 8 is an even number.SO, I know that328 is divisible by 2.

328 ÷ 2 = 164

Is 328 divisible by 3?Use the rule.1. Add the digits of 328

together.2. 3 + 2 + 8 = 13.3. Divide the sum of the

digits by 3.4. 13 ÷ 3 = 4 R1So, I know that 328 is NOT divisible by 3.

So, because 328 is NOT divisible by both 2 and 3,

I know that 328 is NOT divisible by 6.

Can I divide this number by 7?YES

BUT, this is tricky.If1. The number without the last digit attached2. minus the last number times 23. is less than 20 and divisible by 7 the whole number is divisible by 7.4. If the number without the last digit attached5. minus the last number times 26. is more than 20, then do the same thing again over and over again until

the difference is 20 or less. 7. Take the new number (the difference) without the last digit attached,8. subtract the last number times 29. If the difference is less than 20, and the number is divisible by 7, then the

original number is divisible by 7.

Is 329 divisible by 7?

Use the rule.1. Take the number without the last digit attached.2. 323. Subtract the last digit 9 times 2.4. 9 x 2 = 185. 32 – 18 = 14 (difference)6. If the difference is less than 20, and it is divisible

by 7, the original number is divisible by 7.7. 14 is less than 20, 14 is divisible by 7.

14 ÷ 7 = 2So, I know that329 is divisible by 7.

329 ÷ 7 = 47

Is 328 divisible by 7?

Use the rule.1. Take the number without the last digit attached.2. 323. Subtract the last digit 8 x2.4. 8 x 2 = 165. 32 – 16 = 16 (difference)6. If the difference is less than 20 is it divisible by 7,

the original number is divisible by 7.7. 16 is less than 20; 16 is NOT divisible by 7.SO, I know that328 is not divisible by 7.

Can I divide this number by 8?

YES

If it’s last three digits taken together as a number are divisible by 8.

Is 4,328 divisible by 8?

Use the rule.1. Take the last 3 digits.2. 3283. See if they are

divisible by 8 without a remainder.

4. If it is, the whole number is divisible by 8.

5. 328 is divisible by 8.So, I know that 4,328 is divisible by 8.

4,328 ÷ 8 = 541

See, it works.

41 5418)328 8)4328 -32 -40 08 32 - 8 -32 0 08 - 8 0

Can I divide this number by 9?

YES

If the sum of the digits is divisible by 9.

Is 324 or 327 divisible by 9?

• Use the rule.• Add the digits in the

number 324.• 3 + 2 + 4 = 9• Divide the sum of the

digits by 9.• 9 ÷ 9 = 1 R0• If the remainder is 0, the

number is divisible by 9.

So I know that,324 is divisible by 9.

324 ÷ 9 = 36

Use the rule.1. Add the digits in the

number 327.2. 3 + 2 + 7 = 123. Divide the sum of the

digits by 9.4. 12 ÷ 9 = 1 R35. If the remainder is 0,

the number is divisible by 9.

So I know that,327 is not divisible by 9.

327 ÷ 9 = 36 R3

Can I divide this number by 10?

YES

If the last digit in the number is a 0.

Is 328 divisible by 10?

Use the rule.

1. The last digit of 328 is 8.

2. 8 is not 0 (zero).

So, I know that

328 is NOT divisible by 10.

But,

1. The last digit in 320 is 0.

So, I know that

320 is divisible by 10.

And

1. The last digit in 330 is 0.

So, I know that

330 is divisible by 10.

What happens when I divide a number by 0?

Remember fact families.Division

12 ÷ 4 = 3 or 12 = 3 4

Multiplication is the inverse (opposite).

quotient x divisor = dividend

3 x 4 = 12or, working backwards,

12 = 4 x 3

BUT, when I try to divide with 0.

12 ÷ 0 or 12 = undefined not 0 0 quotient

because, when I multiply (do the opposite or inverse).

no quotient x 0 will = 12 and

0 x 0 = 0 not 12

There is no number in the whole world I can put in the quotient’s place as a factor that will equal 12, when I multiply it by 0. The product will always be 0.

So, when I divide a number by 0,the result is called “undefined.”

Congratulations!

Now you know the short cut for determining if a number can be divided by another number!

This will come in handy when you are finding prime and composite numbers, prime factors (prime factorization), greatest common factors (GCF), least common multiples (LCM), and need to find equivalent fractions.

Notes for teachers on texts correlation and design:

Correlates with Glencoe Mathematics (Florida Edition) texts: Mathematics: Applications and Concepts Course 1: (red book) Chapter 1 Lesson 2 Divisibility PatternsMathematics: Applications and Concepts Course 2: (blue book) Prerequisite skillsPre-Algebra: (green book) Chapter 4 Lesson 1: Factors and Monomials

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Thank you for viewing this slide presentation. Thanks to my colleague, Sarah, for sharing much of this information with me. I hope you will find it of help to your students. Taleese

For more information on my math class see http://www.walsh.edublogs.org