Divisibility RulesDivisibility Rules

Outcome A6 – Develop and apply Outcome A6 – Develop and apply divisibility rules for 3, 4, 6, 9divisibility rules for 3, 4, 6, 9

Divisibility rule for 3Divisibility rule for 3

If you add the digits of a number divisible If you add the digits of a number divisible by three the sum must be divisible by by three the sum must be divisible by three.three.

Example: Example: 465 465 → 4 + 6 + 5 = 15→ 4 + 6 + 5 = 1515 → 1 + 5 = 6 15 → 1 + 5 = 6 6 ÷ 3 = 26 ÷ 3 = 2Because 6 was divisible by three 465 is divisible by Because 6 was divisible by three 465 is divisible by

threethree

Divisibility rule for 4Divisibility rule for 4

If a number is divisible by 4 the last two If a number is divisible by 4 the last two digits are divisible by 4.digits are divisible by 4.

Example:Example:1364 1364 → 64 ÷ 4 = 16→ 64 ÷ 4 = 16

**remember this works because 100 is divisible by 4 **remember this works because 100 is divisible by 4 (100 ÷ 4 = 25) that means 1300 is divisible by 4 so (100 ÷ 4 = 25) that means 1300 is divisible by 4 so we only need to worry about the last two digits.we only need to worry about the last two digits.

Divisibility rule for 6Divisibility rule for 6

If a number is divisible by 6 it must be If a number is divisible by 6 it must be divisible by both 2 and 3.divisible by both 2 and 3.

Remember a number is divisible by two if it is Remember a number is divisible by two if it is even. See earlier notes for divisibility of 3.even. See earlier notes for divisibility of 3.

Example:Example:672 672 → is an even number so it is divisible by 2→ is an even number so it is divisible by 2672 → 6 + 7 + 2 = 15 → 1 + 5 = 6 → 6 ÷ 3 = 2672 → 6 + 7 + 2 = 15 → 1 + 5 = 6 → 6 ÷ 3 = 2Because the number meets both rules it is divisible Because the number meets both rules it is divisible

by 6.by 6.

Divisibility rule for 9Divisibility rule for 9

If you add the digits of a number divisible If you add the digits of a number divisible by 9 then the sum of the numbers must be by 9 then the sum of the numbers must be divisible by 9.divisible by 9.

Example: Example: 576 576 → 5 + 7 + 6 = 18 → 18→ 5 + 7 + 6 = 18 → 18 ÷ 9 = 2÷ 9 = 2576 is divisible by 9576 is divisible by 9

Divisibility Questions.Divisibility Questions.

1. Is 136 divisible by 3? 1. Is 136 divisible by 3? Explain why or why not.Explain why or why not.

2. Is 244 divisible by 4. 2. Is 244 divisible by 4. Explain why or why not.Explain why or why not.

3. Is 294 divisible by 6?3. Is 294 divisible by 6?Explain why or why not.Explain why or why not.

4. Is 5652 divisible by 9?4. Is 5652 divisible by 9?Explain why or why not.Explain why or why not.

5. Fill in the missing digit so 5. Fill in the missing digit so the numbers are divisible by the numbers are divisible by

9.9.a.a. 34__34__

b.b. 59__4559__45

c.c. __5004__5004

6. Guess the number…6. Guess the number…

The number is odd and greater The number is odd and greater than 200 but less than 401. The than 200 but less than 401. The number is divisible by 5 and 9.number is divisible by 5 and 9.

7. Guess the number…7. Guess the number…

The number is less than 100 and The number is less than 100 and is divisible by 2, 5 and 8.is divisible by 2, 5 and 8.

8. Guess the number…8. Guess the number…

The number is less than 172 but The number is less than 172 but greater than 90. The number is greater than 90. The number is

divisible by 3, 5, and 9.divisible by 3, 5, and 9.

9. Create your own guess 9. Create your own guess the number question.the number question.

Divisibility Answers.Divisibility Answers.

1. Is 136 divisible by 3? 1. Is 136 divisible by 3? Explain why or why not.Explain why or why not.

No, because 1+3+6 = 10 and 10 is not No, because 1+3+6 = 10 and 10 is not divisible by 3. The divisibility rule for divisible by 3. The divisibility rule for

three is that if we add all the digits, the three is that if we add all the digits, the sum must be divisible by three.sum must be divisible by three.

2. Is 244 divisible by 4. 2. Is 244 divisible by 4. Explain why or why not.Explain why or why not.

Yes, because 44 Yes, because 44 ÷ 4 = 11. The ÷ 4 = 11. The divisibility rule for 4 is that if the last divisibility rule for 4 is that if the last

two digits are divisible by four then the two digits are divisible by four then the whole number is divisible by 4.whole number is divisible by 4.

3. Is 294 divisible by 6?3. Is 294 divisible by 6?Explain why or why not.Explain why or why not.

Yes, because it is even and 2+9+4 = Yes, because it is even and 2+9+4 = 15 and 15 15 and 15 ÷ 3 = 5. That means 294 ÷ 3 = 5. That means 294 meets the divisibility rule for 2 and 3 meets the divisibility rule for 2 and 3

and is therefore divisible by 6.and is therefore divisible by 6.

4. Is 5652 divisible by 9?4. Is 5652 divisible by 9?Explain why or why not.Explain why or why not.

Yes, because 5+6+5+2 = 18 and 18 is Yes, because 5+6+5+2 = 18 and 18 is divisible by 9. Since the sum of the divisible by 9. Since the sum of the

digits is divisible by 9 then the number digits is divisible by 9 then the number is divisible by 9.is divisible by 9.

5. Fill in the missing digit so 5. Fill in the missing digit so the numbers are divisible by the numbers are divisible by

9.9.a.a. 34__ = 34234__ = 342

b.b. 59__45 = 5944559__45 = 59445

c.c. __5004 = 95004 or 05004__5004 = 95004 or 05004

6. Guess the number… The number is 6. Guess the number… The number is odd and greater than 200 but less than odd and greater than 200 but less than 401. The number is divisible by 5 and 9.401. The number is divisible by 5 and 9.

If the number is divisible by 5 and odd the last digit must be a 5. the If the number is divisible by 5 and odd the last digit must be a 5. the three numbers must add up to a multiple of 9. Since the last digit is a 5 three numbers must add up to a multiple of 9. Since the last digit is a 5

the first two digits must add up to 4 or 13. They can’t add up to 13 the first two digits must add up to 4 or 13. They can’t add up to 13 because the first digit must be either 2 or 3.because the first digit must be either 2 or 3.

If the first digit is a 2 and the last digit is a 5 then the number must be If the first digit is a 2 and the last digit is a 5 then the number must be 225 (2+2+5=9)225 (2+2+5=9)

If the first digit is a 3 then it must be 3+1+5=9 so 315If the first digit is a 3 then it must be 3+1+5=9 so 315

So there are two answers 225 and 315So there are two answers 225 and 315

7. Guess the number… The 7. Guess the number… The number is less than 100 and is number is less than 100 and is

divisible by 2, 5 and 8.divisible by 2, 5 and 8.Because the number is divisible by both 2 and 5 the Because the number is divisible by both 2 and 5 the

number must end in a zero (5 must have a last digit of 5 number must end in a zero (5 must have a last digit of 5 or 0 but 5 is not even so it is not divisible by 2) so the or 0 but 5 is not even so it is not divisible by 2) so the number could be 10, 20, 30, 40, 50, 60, 70, 80, or 90number could be 10, 20, 30, 40, 50, 60, 70, 80, or 90

The number must also be divisible by 8 so it could be The number must also be divisible by 8 so it could be either 40 or 80 as they are both multiples of 8.either 40 or 80 as they are both multiples of 8.

8. Guess the number…The number is less than 8. Guess the number…The number is less than 172 but greater than 90. The number is divisible 172 but greater than 90. The number is divisible

by 3, 5, and 9.by 3, 5, and 9.

We know that all numbers that are divisible by 9 are also We know that all numbers that are divisible by 9 are also divisible by 3 so we only have to worry about 9 and 5. divisible by 3 so we only have to worry about 9 and 5.

From the divisibility rule for 5 we know the number must end From the divisibility rule for 5 we know the number must end in a 5 or 0. From the divisibility rule for 9 we know the sum of in a 5 or 0. From the divisibility rule for 9 we know the sum of

the digits must be divisible by 9. the digits must be divisible by 9. If we guess the last digit is 0 then the first digit or first two If we guess the last digit is 0 then the first digit or first two digits must add to 9. 90 does not work because the number digits must add to 9. 90 does not work because the number

must be greater than 90. and the greatest number we can go must be greater than 90. and the greatest number we can go to is 170 but 1+7+0 is only 8 so we do not have a factor of 9. to is 170 but 1+7+0 is only 8 so we do not have a factor of 9.

So the last number must be 5.So the last number must be 5.If the last number is 5 then the first digit or digits must add If the last number is 5 then the first digit or digits must add

together with 5 to equal 9. The first digit must be a 1.together with 5 to equal 9. The first digit must be a 1.So 1+___+5=9 the second digit must be a 3So 1+___+5=9 the second digit must be a 3

135 is divisible by 3, 5, and 9.135 is divisible by 3, 5, and 9.

9. Create your own guess 9. Create your own guess the number question.the number question.

PracticePractice

No Calculators allowed!!No Calculators allowed!!Textbook page 31-33Textbook page 31-33

#1-8, 11-13, 15, 18, 19#1-8, 11-13, 15, 18, 19