What do we mean by “x is divisible by y”? It means – If we divide x by y, the result is a whole number and the remainder is 0. So, if I say 99 is divisible by 9; it means the remainder of 99 ÷ 9 is 0.
Examples:
- 108 is divisible by 12 because 108 ÷ 12 = 9
- 75 is divisible by 3 because 75 ÷ 3 = 25
- 66 is not divisible by 4 because 66 ÷ 4 leaves a remainder of 2.
- 0 is divisible by 9 because 0 ÷ 7 = 0
What is a divisibility rule?
Divisibility rules are shortcuts for determining whether a number is divisible by another; without actually performing the calculation.
Let us look at the divisibility rules of various numbers:
What is the divisibility rule for 1?
Every number is divisible by 1, so there is no specific rule for this 🙂
x ÷ 1 = x for every x.
What is the divisibility rule for 2?
If the last digit of the number is 0, 2, 4, 6 or 8; the number is divisible by 2.
Examples:
| Number | Last digit | Is Divisible as per rule? | Remainder after actual calculation |
| 128 | 8 | Yes | 0 |
| 9875 | 5 | No | 1 |
| 876578453 | 3 | No | 1 |
What is the divisibility rule for 3?
If the sum of all the digits of the number is divisible by 3, the number is divisible by 3. This rule can be used repetitively till we get a single digit number.
Examples:
| Number | Sum of the digits | Is Divisible as per rule? | Remainder after actual calculation |
| 87 | 15 | Yes | 0 |
| 9873 | 9 | Yes | 0 |
| 876578453 | 8 | No | 2 |
What is the divisibility rule for 4?
If the number formed by the last 2 digits of the given number is divisible by 4, the given number is divisible by 4.
Examples:
| Number | Number formed by the last 2 digits | Number formed by the last 2 digits divisible by 4? | Number divisible as per the rule? | Remainder after actual calculation |
| 87 | 87 | No | No | 3 |
| 9876544 | 44 | Yes | Yes | 0 |
| 876578453 | 53 | No | No | 1 |
| 98754304 | 4 | Yes | Yes | 0 |
What is the divisibility rule for 5?
If the last digit of the number is either 5 or 0, the number is divisible by 5.
Examples:
| Number | Last digit | Number divisible as per the rule? | Remainder after actual calculation |
| 87 | 7 | No | 2 |
| 9876530 | 0 | Yes | 0 |
| 876578455 | 5 | Yes | 0 |
| 98754304 | 4 | No | 4 |
What is the divisibility rule for 6?
If the number is divisible by both 2 and 3, the number is divisible by 6.
Examples:
| Number | Last digit | Divisible by 2? | Sum of the numbers | Divisible by 3? | Divisible by 6 as per the rule? | Remainder after actual calculation |
| 87 | 7 | No | Does not matter | No | 3 | |
| 9876530 | 0 | Yes | 2 | No | No | 5 |
| 8765784516 | 6 | Yes | 12 | Yes | Yes | 0 |
| 987543042 | 2 | Yes | 6 | Yes | Yes | 0 |
Note that in the first example, we need not find out if the number is divisible by 3. If it is not divisible by 2, it will not be divisible by 6 even if it is divisible by 3.
What is the divisibility rule for 7?
Double the last digit and subtract it from the number formed by the rest of the digits. If the result is divisible by 7, the original number is divisible by 7. This rule can be repeated over and over.
Examples:
| Number | Double of the last digit | Number formed by the rest of the digits | Result of the subtraction | Result obtained by doing this repetitively | Result divisible by 7? | Divisible by 7 as per the rule? | Remainder after actual calculation |
| 872 | 4 | 87 | 87 – 4 = 83 | No | No | 4 | |
| 9876530 | 0 | 987653 | 987653 – 0 = 987653 | 95 | No | No | 6 |
| 8950578 | 16 | 895057 | 895057 – 16 = 895041 | 0 | Yes | Yes | 0 |
| 68355 | 10 | 6835 | 6835 – 10 = 6825 | 0 | Yes | Yes | 0 |
What is the divisibility rule for 8?
If the number formed by the last 3 digits of the given number is divisible by 8, the given number is divisible by 8.
Examples:
| Number | Number formed by the last 3 digits | Number formed by last 3 digits divisible by 8? | Divisible by 8 as per the rule? | Remainder after actual calculation |
| 872 | 872 | Yes | Yes | 0 |
| 9876530 | 530 | No | No | 2 |
| 8950576 | 576 | Yes | Yes | 0 |
| 68355 | 355 | No | No | 3 |
A quick method that I apply to check the divisibility of a number by 8 is to halve the number 2 times. If the result is a whole number, the number is divisible by 8. For eg. 872 /2 = 436 and 436 / 2 = 218 which is a whole number.
What is the divisibility rule for 9?
If the sum of the digits adds up to 9, the given number is divisible by 9. This rule can be repeated till we get a single digit number.
Examples:
| Number | Sum of the digits | Repeat the process till we get single digit | Is the result 9? | Divisible by 9 as per the rule? | Remainder after actual calculation |
| 872 | 17 | 8 | No | No | 8 |
| 9876530 | 38 | 2 | No | No | 2 |
| 5106888 | 36 | 9 | Yes | Yes | 0 |
| 68355 | 27 | 9 | Yes | Yes | 0 |
What is the divisibility rule for 10?
If the last digit of the number is 0, the given number is divisible by 10. It can’t get easier than that 🙂
| Number | Last digit | Divisible by 10 as per the rule? | Remainder after actual calculation |
| 872 | 2 | No | 2 |
| 9876530 | 0 | Yes | 0 |
What is the divisibility rule for 11?
Add the digits in odd position. The add the digits in even position. If the difference between the 2 results is divisible by 11, the original number is divisible by 11.
| Number | Sum of the digits in odd position | Sum of the digits in even position | Difference between the 2 sums | Is the difference divisible by 11? | Divisible by 11 as per the rule? | Remainder after actual calculation |
| 8709 | 8 + 0 = 8 | 7 + 9 = 16 | 16 – 8 = 8 | No | No | 8 |
| 9876530 | 9 + 7 + 5 + 0 = 21 | 8 + 6 + 3 = 17 | 21 – 17 = 4 | No | No | 4 |
| 96419818 | 9 + 4 + 9 + 1 = 23 | 6 + 1 + 8 + 8 = 23 | 23 – 23 = 0 | Yes | Yes | 0 |
What is the divisibility rule for 12?
If the number is divisible by both 3 and 4, it is divisible by 12.
Examples:
| Number | Sum of the digits | Is the number divisible by 3? | Number formed by last 2 digits | Is the number divisible by 4? | Divisible by 12 as per the rule? | Remainder after actual calculation |
| 8709 | 24 | Yes | 9 | No | No | 9 |
| 9876530 | 11 | No | Does not matter | No | 2 | |
| 1209180 | 21 | Yes | 80 | Yes | Yes | 0 |
Note that in the second examples, we need not find out of the number is divisible by 4. Since it is not divisible by 3, it does not matter if it is divisible by 4.
What is the divisibility rule for 13?
Remember the sequence 1, 10, 9, 12, 3, 4. Multiply the right most digit of the number with the left most number in the sequence, the second right most digit to the second left most digit of the number in the sequence. The cycle goes on and repeats after 5 digits. Add the results of these multiplications. If the sum is divisible by 13, the number is divisible by 13.
Examples:
| Sequence | 1, 10, 9, 12, 3, 4 | |||
| Number | Sum as per the rule | Is the sum divisible by 13? | Divisible by 13 as per the rule? | Remainder after actual calculation |
| 8709 | 9 x 1 + 0 x 10 + 7 x 9 + 8 x 12 = 9 + 0 + 63 + 96 = 168 | No | No | 12 |
| 7654321 | 1 x 1 + 2 x 10 + 3 x 9 + 4 x 12 + 5 x 3 + 6 x 4 + 7 x 1 = 1 + 20 + 27 + 48 + 15 + 24 + 7 = 142 | No | No | 12 |
| 1586 | 6 x 1 + 8 x 10 + 5 x 9 + 1 x 12 = 6 + 80 + 45 + 12 = 143 | Yes | Yes | 0 |
What is the divisibility rule for 14?
If the number is divisible by both 2 and 7, the number is divisible by 14.
Examples:
| Number | Is the number divisible by 2? | Is the number divisible by 7? | Divisible by 14 as per the rule? | Remainder after actual calculation |
| 8709 | No | Does not matter | No | 1 |
| 7654321 | No | Does not matter | No | 11 |
| 1246 | Yes | Yes | Yes | 0 |
What is the divisibility rule for 15?
If the number divisible by both 3 and 5, it is divisible by 15.
| Number | Is the number divisible by 3? | Is the number divisible by 5? | Divisible by 15 as per the rule? | Remainder after actual calculation |
| 8709 | Yes | No | No | 9 |
| 7654321 | No | Does not matter | No | 1 |
| 23505 | Yes | Yes | Yes | 0 |
There are divisibility rules for other numbers which we will discuss in another blog post. Stay tuned…
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