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Divisibility rules include short tricks that determine if a number (Dividend) is exactly divisible by another number (Divisor) or not, without carrying out the actual process of division. Divisibility Rules are also called divisibility tests. If a number is completely divisible by other number, the quotient should be a whole number and remainder should be zero. An example of divisibility rules: 6 is divisible by 3 ( "3 divides 6") because 6/3 = 2, and 2 is a whole number.
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Key Terms: Divisibility Rules, divisibility rule of 7, divisibility rule of 11, divisibility rule of 4, divisibility rule of 8, divisibility
Divisibility Rule/ Test Table
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Refer to below table to understand different divisibility rules:
| Numbers | Divisibility Rule |
|---|---|
| Divisibility by 1 | No specific rule. |
| Divisibility by 2 | When the last digit is divisible by 2. Example: 0, 2, 4, 6, 8, 10, 12, 14, etc. |
| Divisibility by 3 | When the sum of numbers in the dividend is divisible by 3. Example: 3, 6, 9, 12, 15, 18, 21, etc. |
| Divisibility by 4 | When the last two digits of any dividend are divisible by 4. NOTE: Numbers having 00 as their last digits are also divisible by 4. Example: 4, 8, 12, 16, 780, 70744, etc. |
| Divisibility by 5 | When the last digit is either 0 or 5. Example: 25, 35, 45, 95, 105, 200, etc. |
| Divisibility by 6 | When the number is divisible by both 2 and 3. Example: 12, 18, 60, etc. |
| Divisibility by 7 | When the last digit is subtracted twice from the remaining digits and gives the multiple of 7. (Pictorial representation below this table.) Example: 77, 42, 49, 70, etc. |
| Divisibility by 8 | When the last three digits are 000 OR are divisible by 8. Example: 2000, 880, 805256, etc. |
| Divisibility by 9 | When the sum of all digits is divisible by 9. Example: 171, 99, 18, etc. |
| Divisibility by 10 | When the last digit is 0. Example: 12120, 110, 520, 440, etc. |
| Divisibility by 11 | When the difference of the sums of the alternative digits is divisible by 11, the number is divisible by 11. Example: 1122, 814, 592845, etc. |
| Divisibility by 12 | When a number is both divisible by 3 and 4. |
| Divisibility by 13 | Multiply 4 to the last digit and add this new number to the remaining given dividend. Continue the process till a two-digit number is found. If the two-digit number is divisible by 13, the dividend is divisible. |
| Divisibility by 17 | Multiply 5 to the last digit and subtract this number from the remaining given dividend. If the result is divisible by 17, the given dividend is as well. |
| Divisibility by 19 | Multiply 2 to the last digit and add this number to the remaining given dividend. If the result is divisible by 19, the given dividend is also divisible by 19. |
Things to Remember
- A number is divisible by 2 if its units place is either 0 or multiple of 2.
- A number is divisible by 3 if the sum of digits is a multiple of 3.
- A number is divisible by 4, if the number formed by its last 2 digits is divisible by 4.
- A number is divisible by 5 if its units place is 0 or 5.
- A number is divisible by 6 if it is divisible by 2 and 3 both.
- We need to double the last digit of the number and then subtract it from the remaining number. If the result is divisible by 7, then the original number will also be divisible by 7.
- A number is divisible by 8 if the number formed by the last 3 digits is divisible by 8.
- A number is divisible by 9 if the sum of its digits is divisible by 9.
- A number is divisible by 10 if it has zero (0) in its units place.
- A number is divisible by 11 if the sum of the digits in the odd places and the sum of the digits in the even places difference is a multiple of 11 or zero.
- A number is divisible by 12, if it is divisible by co-prime 12 i.e 3 and 4.
- A number is divisible by 15, if it is divisible by co-prime 15 i.e., 3 and 5.
- A number is divisible by 18, if it is divisible by co-prime 18 i.e., 2 and 9.
- A number is divisible by 45, if it is divisible by co-prime 45 i.e., 5 and 9.
Sample Questions
Ques. Use the divisibility tests to determine whether 99 is divisible by 2, 3, 4, 5, 6, 9 or 11.
Ans. The sum of the digits 9 and 9 is 18 which is divisible by both 3 and 9.
If the sum of all the digits in a number is divisible by 3 and 9, then the number is divisible by both 3 and 9. So, the number 99 is divisible by both 3 and 9.
Starting from the right-hand side, the digit at odd places is 9 and the digit at even place is 9.
Difference between the two is 9 – 9 = 0
If the difference between the sums obtained by adding alternate digits of a number is 0 or divisible by 11, then the number is divisible by 11. Therefore, the number 99 is divisible by 11 also.
In all, the number 99 is divisible by 3, 9 and 11.
Ques. Use the divisibility tests to determine whether 135 is divisible by 2, 3, 4, 5, 6, 9 or 11.
Ans. We observe that the sum of the digits 1, 3 and 5 gives us the result 9, which is divisible by both 3 and 9 and if the sum of all the digits in a number is divisible by 3 and 9, then the number is divisible by both 3 and 9. So, the number 135 is divisible by 3 and 9.
Also, the number 135 has its last digit as 5.
Considering the fact that if a number has either 0 or 5 in its unit’s place, then the number is divisible by 5. Therefore, the number 135 is divisible by 5.
In all, the number 135 is divisible by 3, 5 and 9.
Ques. Use the divisibility tests to determine whether 711 is divisible by 2, 3, 4, 5, 6, 9 or 11.
Ans. The sum of the digits 7, 1 and 1 gives us 9, which is divisible by both 3 and 9.
If the sum of all the digits in a number is divisible by 3 and 9, then the number is divisible by 3 and 9. So, the number 711 is divisible by both 3 and 9.
Ques. Use the divisibility tests to determine whether 280 is divisible by 2, 3, 4, 5, 6, 9 or 11.
Ans. We observe here that the number has its last digit as 0. If a number has either 0 or 5 as its last digit, then the number is divisible by 5. Also, if a number has the digits 0, 2, 4, 6 or 8 as its last digit, then the number is divisible by 2. So, the number 280 is divisible by both 2 and 5.
The number formed by the last two digits is 80, which is divisible by 4.
If the number formed by the digits in the tens and units places of a number is divisible by 4, then the number is divisible by 4. So, the number 280 is divisible by 4.
In all, the number 280 is divisible by 2, 4 and 5.
Ques. Use the divisibility tests to determine whether 378 is divisible by 2, 3, 4, 5, 6, 9 or 11.
Ans. The number has the digit 8 in its unit’s place.
Considering that a number has 0, 2, 4, 6 or 8 in its unit’s place then the number is divisible by 2, the number 378 is divisible by 2.
The sum of the digits 3, 7 and 8 is 18, which is divisible by 3 and 9.
If the sum of all the digits in a number is divisible by 3 and 9, then the number is divisible by both 3 and 9. So, the number 378 is divisible by 3 and 9.
Since the number is divisible by both 2 and 3, it is also divisible by 6.
In all, the number 378 is divisible by 2, 3, 6 and 9.
Ques. Use the divisibility tests to determine whether 495 is divisible by 2, 3, 4, 5, 6, 9 or 11.
Ans. Observing that the sum of the digits is 4, 9 and 5 is 18, which is divisible by both 3 and 9.
If the sum of all the digits in a number is divisible by 3 and 9, then the number is divisible by both 3 and 9. Thus, the number 495 is divisible by both 3 and 9.
The number has the digit 5 in its unit’s place.
If a number has either 0 or 5 in its unit’s place, then the number is divisible by 5. So, the number 495 is divisible by 5.
Starting from the right-hand side, the digits at odd places are 5 and 4.
Sum = 5 + 4 = 9
The digit at even place is 9.
Difference between both = 9 – 9 = 0
If the difference between the sums obtained by adding alternate digits of a number is 0 or divisible by 11, then the number is divisible by 11. Thus, the number 495 is divisible by 11.
In all, the number 495 is divisible by 3, 5, 9 and 11.
Ques. Use the divisibility tests to determine whether 504 is divisible by 2, 3, 4, 5, 6, 9 or 11.
Ans. Observing that the digit in the unit’s place is 4.
If a number has 0, 2, 4, 6 or 8 in its unit’s place, then the number is divisible by 2. So, the number 504 is divisible by 2.
The sum of the digits 5, 0 and 4 is 9, which is divisible by both 3 and 9.
If the sum of all the digits in a number is divisible by 3 and 9, then the number is divisible by both 3 and 9. Therefore, the number 504 is divisible by both 3 and 9.
The number formed by the digits in the tens and unit’s places is 04, which is divisible by 4.
If the number formed by the digits in the tens and units places of a number is divisible by 4, then the number is divisible by 4 so the number 504 is divisible by 4.
If a number is divisible by both 2 and 3, then the number is divisible by 6. So, the number 504 is also divisible by 6.
In all, the number 504 is divisible by 2, 3, 4, 6 and 9.
Ques. Use the divisibility tests to determine whether 616 is divisible by 2, 3, 4, 5, 6, 9 or 11.
The digit in the unit’s place is 6.
If a number has 0, 2, 4, 6 or 8 in its unit’s place, then the number is divisible by 2. Subsequently, the number 616 is divisible by 2.
The number formed by the digits in the tens and units places is 16, which is divisible by 4.
If the number formed by the digits in the tens and units places of a number is divisible by 4, then the number is divisible by 4. Thus, the number 616 is divisible by 4.
Starting from the right-hand side, the digits at odd places are 6 and 6.
Sum = 6 + 6 = 12
The digit at even place is 1.
Difference between both of them = 12 – 1 = 11
If the difference between the sums obtained by adding alternate digits of a number is 0 or divisible by 11, then the number is divisible by 11. So, the number 616 is divisible by 11.
In all, the number 616 is divisible by 2, 4 and 11.
Ques. Use the divisibility tests to determine whether 720 is divisible by 2, 3, 4, 5, 6, 9 or 11.
Ans. The digit in the unit’s place is 0.
If a number has either 0 or 5 in its unit’s place, then it is divisible by 5 and if a number has 0, 2, 4, 6 or 8 in its unit’s place, then it is divisible by 2. So, the number 720 is divisible by both 2 and 5.
The sum of the digits 7, 2 and 0 is 9, which is divisible by both 3 and 9.
If the sum of all the digits in a number is divisible by 3 and 9, then the number is divisible by 3 and 9. Thus, the number 720 is divisible by both 3 and 9.
If a number is divisible by both 2 and 3, then the number is divisible by 6 which means the number 720 is divisible by 6.
The number formed by the digits in the tens and units places is 20, which is divisible by 4.
If the number formed by the digits in the tens and units places of a number is divisible by 4, then the number is divisible by 4. Thus, the number 720 is divisible by 4.
In all, the number 720 is divisible by 2, 3, 4, 5, 6 and 9.
Ques. Use the divisibility tests to determine whether 2,304 is divisible by 2, 3, 4, 5, 6, 9 or 11.
Ans. The digit in the unit’s place is 4.
If a number has 0, 2, 4, 6 or 8 in its unit’s place, then the number is divisible by 2. Therefore, the number 2304 is divisible by 2.
The sum of the digits 2, 3, 0 and 4 is 9, which is divisible by both 3 and 9.
If the sum of all the digits in a number is divisible by 3 and 9, then the number is divisible by 3 and 9. Thus, the number 2304 is divisible by both 3 and 9.
If a number is divisible by the numbers 2 and 3, then the number is divisible by 6. Subsequently, the number 2304 is divisible by 6.
The number formed by the digits in the tens and units places is 04, which is divisible by 4.
If the number formed by the digits in the tens and units places of a number is divisible by 4, then the number is divisible by 4. Therefore, the number 2304 is divisible by 4.
In all, the number 2304 is divisible by 2, 3, 4, 6 and 9.
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