A Divisibility Rule is like a shortcut to determine if a particular number can be divided to another number without any remainder. This can be a useful trick when looking for factors of a given number.
A long method of finding out divisibility is to divide the number and check if there is a remainder. This is accurate but if the numbers are big, it will take time before you will arrive at an answer.
The divisibility rules works like a shortcut. It only needs a few numbers or a few simple operations to check for divisibility. The divisibility rules are usually for numbers 2 to 10. There are also divisibility rules for 11 and 12 but it is not included here since they are not used that often.
Below is a summary of the Divisibility Rules for 2, 3, 4, 5, 6, 7, 8, 9 and 10. A detailed process and examples are provided below for each divisibility test.
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Divisibility Test for 2: If the last digit is 2, 4, 6, 8, 0. Or as long as it is an even number, it is divisible by 2.
Examples:
- 48 is divisible by 2 since the last digit is 8.
- 2,574 is divisible by 2 since the last digit is 4.
- 57, 330 is divisible by 2 since the last digit is 0.
As you can see, you don’t need to perform a long division to determine the divisibility, just checking 1 digit is enough.
Divisibility Test for 3: If the sum of all the digits is divisible by 3.
Examples:
- 48 The sum of the digits is 4 + 8 = 12. The sum, which is 12, is divisible by 3. Thus, 48 is divisible by 3.
- 192 The sum of the digits is 1 + 9 + 2 = 12. Again, the sum is divisible by 3. Thus, 192 is divisible by 3.
- 100,200 The sum of the digits is 1+0+0+2+0+0 = 3. The sum of the digits is divisible by 3. Thus, 100,200 is divisible by 3.
Divisibility Test for 4: If the last two digits is divisible by 4.
Examples:
- 544 The last two digits is 44. This is divisible by 4. Thus, 544 is divisible by 4 also.
- 1,208 The last two digits is 08. This is divisible by 4. Thus, 1,208 is divisible by 4.
- 1,053,312 The last two digits is 12. This is divisible by 4. Thus, 1,053,312 is divisible by 4.
Divisibility Test for 5. If the last digit is 5 or 0.
Examples:
- 345 The last digit is 5, thus 345 is divisible by 5.
- 10,870 The last digit is 0, thus 10,870 is divisible by 5.
- 1,562,095 The last digit is 5, thus it is divisible by 5.
Divisibility Test for 6. If the number is divisible by BOTH 2 and 3.
Examples:
- 72 This number is divisible by 2 since it is an even number. At the same time, the sum of the digits ( 7 + 2 = 9 ) is divisible by 3. Thus, 72 is divisible by 6.
- 624 This number is divisible by 2 since it is an even number. At the same time, the sum of the digits ( 6 + 2 + 4 = 12 ) is divisible by 3. Thus, 624 is divisible by 6.
- 112,224 This number is divisible by 2 since it is an even number. The sum of the digits ( 1 + 1 + 2 + 2 + 2 + 4 = 12 ) is divisible by 2. Therefore, 112,224 is divisible by 6.
Divisibility Test for 7. Double the last digit, then subtract it from the rest of the number. Repeat if needed. The difference should be divisible by 7 (including 0).
Examples:
- 672 First, double the last digit as 2 x 2 = 4. Then subtract 4 to the remaining digits as 67 – 4 = 63. Since 63 is divisible by 7, the number 672 is divisible by 7.
- 2,114 First, double the last digit as 4 x 2 = 8. Then subtract 8 to the remaining digits as 211 – 8 = 203. We can repeat the steps again using 203. Double the last digit as 3 x 2 = 6. Then subtract 6 to the remaining digits as 20 – 6 = 14. Here, it is not clear that 14 is divisible by 7. Therefore, the number 2,114 is divisible by 7.
Divisibility Test for 7. Double the last digit, then subtract it from the rest of the number. Repeat if needed. The difference should be divisible by 7 (including 0).
Examples:
- 672 First, double the last digit as 2 x 2 = 4. Then subtract 4 to the remaining digits as 67 – 4 = 63. Since 63 is divisible by 7, the number 672 is divisible by 7.
- 2,114 First, double the last digit as 4 x 2 = 8. Then subtract 8 to the remaining digits as 211 – 8 = 203. We can repeat the steps again using 203. Double the last digit as 3 x 2 = 6. Then subtract 6 to the remaining digits as 20 – 6 = 14. Here, it is not clear that 14 is divisible by 7. Therefore, the number 2,114 is divisible by 7.
Divisibility Test for 8. If the last three digits is divisible by 8.
Examples:
- 2,816 By performing a long division on the last three digits, 816 is divisible by 8. Thus, the number 2,816 is divisible by 8.
- 77,184 By performing a long division on the last three digits, 184 is divisible by 8. Thus, the number 77,184 is divisible by 8.
- 156,016 The last three digits, 016, is divisible by 8. Thus, the number 156,106 is divisible by 8.
Divisibility Test for 9. If the sum of all the digits is divisible by 9.
Examples:
- 882 The sum of the digits is 8 + 8 +2 = 18 which is divisible by 9. Thus 882 is divisible by 9.
- 4,518 The sum of the digits is 4 + 5+ 1 + 8 = 18 which is divisible by 9. Thus, 4,518 is divisible by 9.
- 307,827 The sum of the digits is 3 + 0 + 7 + 8 + 2 + 7 = 27 which is divisible by 9. Thus, 307,827 is divisible by 9.
Divisibility Test for 10. If the last digit is 0.
Examples:
- 140 The last digit is 0, therefore 104 is divisible by 10.
- 5,980 The last digit is 0, therefore 5,980 is divisible by 10.
- 783,210 The last digit is 0, therefore 783,210 is divisible by 10.
