Divisibility Tests

What are divisibility rules?

• Rules that tell us whether a larger number is divisible by smaller prime/ composite number (leaving no remainder) without actually doing the tedious and long division.

Divisibility test

for 2

• If the last digit is divisible by 2, then the number is divisible by 2.

• Example:1346225

5 is not a multiple of 2 therefore 1346225 is not divisible by 2.

• Example:1976246

6 is a multiple of 2 therefore 1976246 is divisible by 2.

Divisibility test

for 4

• If the last two digits are divisible by 4, then the number is divisible by 4.

• Example: 1346225

25 is not a multiple of 4 therefore 1346225 is not divisible by 4.

• Example: 1976244

44 is a multiple of 4 therefore 1876244 is divisible by 4.

Divisibility test

for 8

• If the last three digits are divisible by 8, then the number is divisible by 8.

• Example: 1346225

225 is not a multiple of 8 therefore 1346225 is not divisible by 8.

• Example: 1976240

240 is a multiple of 8 therefore 1876240 is divisible by 8.

Divisibility test

for 16

• If the last four digits are divisible by 16, then the number is divisible by 16.

• Example: 1346225

6225 is not a multiple of 16 therefore 1346225 is not divisible by 16.

• Example: 1976240

6240 is a multiple of 16 therefore 1876240 is divisible by 16.

Do you notice a pattern?

• Divisibility test for 4 last two digits

• Divisibility test for 8 last three digits

• Divisibility test for 16 last four digits

• Why is it sufficient to only consider the last few digits?

Explanation

• Since 100 is a multiple of 4, it follows that a number is divisible by 4 if the number formed by its last two digits is a multiple of 4

• Since 1000 is a multiple of 8, it follows that a number is divisible by 8 if the number formed by its last three digits is a multiple of 8

• By similar reasoning, since 10000 is a multiple of 16, a number is divisible by 18 if the number formed by its last four digits is a multiple of 16

Divisibility test

for 5

• If the last digit is either 0 or 5, then the number is divisible by 5.

• Example: 1346225

Last digit is a multiple of 5 therefore 1346225 is divisible by 5.

• Example: 1976247

7 is a not multiple of 5 therefore 1976247 is divisible by 5.

Divisibility test

for 10

• If the last digit is 0, then the number is divisible by 10.

• Example: 1346225

Last digit is a multiple of 5 therefore 1346225 is not divisible by 10.

• Example: 1976240

Last digit is 0 therefore 1976240 is divisible by 10.

Divisibility test

for 3

• If the sum of the digits are divisible by 3, then the number is divisible by 3.

• Example:1346225

Rule: 1 + 3 + 4 + 6 + 2 + 2 + 5 = 23

23 is not a multiple of 3 therefore 1346225 is not divisible by 3.

• Example:1976247

Rule: 1 + 9 + 7 + 6 + 2 + 4 + 7 = 36

36 is a multiple of 3 therefore 1976247 is divisible by 3.

Divisibility test

for 6

• To check if a number is divisible by 6, you need to determine if it is divisible by 2 and 3

• First, determine if the number is divisible by 2

• Next, determine if the sum of the digits are divisible by 3, then the number is divisible by 3.

• Example:1976250

Rule: 1 + 9 + 7 + 6 + 2 + 5 + 0 = 30

30 is a multiple of 3, therefore 1976250 is divisible by 3 and hence, 6.

Divisibility test

for 9

• If the sum of the digits are divisible by 9, then the number is divisible by 9.

Example: 2930485Rule: 2 + 9 + 3 + 0 + 4 + 8 + 5 = 3131 is not a multiple of 9, therefore 2930485 is not divisible by

9.

Example: 8692866Rule: 8 + 6 + 9 + 2 + 8 + 6 + 6 = 4545 is a multiple of 9, therefore 8692866 is divisible by 9.

Divisibility test

for 11

• The difference between the sum of the digits in the odd places and the sum of the digits in the even places is equal to 0 or is a multiple of 11.

• Example: 108394

Sum of digits in even places: 9 + 8 + 1 = 18Sum of digits in odd places: 4 + 3 + 0 = 7Difference = 18 – 7 = 11

The difference is divisible by 11 and therefore the original number is divisible by 11.

Divisibility test

7

• Multiply the left-hand digit by 3 and add the next digit. Multiply the answer by 3. Repeat as often as necessary. If the final answer is divisible by 7, so is the original number.

• Whenever a result is 7 or more, subtract the highest multiple of 7 less than or equal the result, before going on to the next step

• Example: 293863 6,6 15,15 1

1 3 3,3 6,

6 3 18,18 4,4 12,12

2 9

3

5,

5 3 15,1

14

14 7

145 1,1 7

8

6

Divisibility test

larger numbers

• If a positive whole number N is divisible by two coprime numbers a and b, then it is divisible by a x b

• Example:

• If a number is divisible by 5 and 9, then it is divisible by 45 as well, since 5 and 9 are coprime

• If a number is divisible by 6 and 9, it may not be divisible by 54, as 6 and 9 are not coprime.

Coprime? Huh?

• Two positive whole numbers are coprime (or relatively prime) if they have no common positive factors except 1.

Are we coprime?

• 4, 5

• Yes

• 3, 9

• No

• 4, 9

• Yes

• 15, 51

• No