Divisibility Test Trick for Quick Calculations

Note: In every divisibility test, please assume that 0 is divisible by all divisors.

Divisibility Test for 2

A number is divisible by 2 if the last digit is divisible by 2 i.e. last digit is 0, 2, 4, 6 or 8.
E.g. 864 is divisible by 2 since the last digit 4 is divisible by 2.

Divisibility Test for 3

A number is divisible by 3 if the sum of the digits of the number is divisible by 3.
E.g. 861 is divisible by 3 since the sum of the digits is 15 (8 + 6 + 1 = 15), and 15 is divisible by 3.

Divisibility Test for 4

A number is divisible by 4 if the number formed by the last two digits is divisible by 4.
E.g. 504 is divisible by 4 since 04 is divisible by 4.

Divisibility Test for 5

A number is divisible by 5 if the last digit is either 0 or 5.
E.g. 425 is divisible by 5 since the last digit is 5.

Divisibility Test for 6

A number is divisible by 6 if it is divisible by 2 AND it is divisible by 3.
E.g. 186 is divisible by 6 since it is divisible by 2 AND it is divisible by 3.

Divisibility Test for 7

To find out if a number is divisible by 7 or not, follow these steps:
1. Separate the last digit from the rest of the number. Let us call the rest of the number the truncated number. The truncated number has one less digit than the original number or the previous truncated number.
2. Double the last digit and subtract it from the truncated number.
3. Check if this result is sufficiently small so that you can immediately say if this is divisible by 7. If it is divisible by 7, then so was the original number. If it is not divisible by 7, then neither was the original number.
4. If the number is still too large to visually check if it is divisible, apply this rule over and over again as necessary.

E.g. Check 6132. The last digit is 2 and the truncated number is 613. Twice of 2 is 4. So subtract 4 from the truncated number 613 i.e. 613 – 4 = 609. Again, the last digit is now 9, and the truncated number is 60. Twice of 9 is 18. Subtract it from the truncated number 60, i.e. 60 – 18 = 42. Now 42 is small enough to check visually. We know that 42 is divisible by 7, so we can tell that 6132 is divisible by 7 also.

NOTE: Explaining this step is long. But actually using it is a very short and time saving method. This method is especially useful in Geometry and Mensuration problems where the value of π plays an important role.

Divisibility Test for 8

A number is divisible by 8 if the number formed by the last three digits is divisible by 8.
E.g. 8120 is divisible by 8 since 120 is divisible by 8.

Divisibility Test for 9

A number is divisible by 9 if the sum of its digits is divisible by 9. If the sum is large, you can once again add the digits and check if the new sum is divisible by 9.
E.g. 27549 is divisible by 9 since the sum of the digits is 27 (2 + 7 + 5 + 4 + 9 = 27), and 27 is divisible by 9.

Divisibility Test for 10

A number is divisible by 10 if the last digit is 0.
E.g. 1760 is divisible by 10 since the last digit is 0.

Divisibility Test for 11

Method 1:- The (sum of the odd place digits) – (sum of the even place digits) is divisible by 11.

E.g. 963391
Sum of odd place digits = 9 + 3 + 9 = 21
Sum of even place digits = 6 + 3 + 1 = 10
Difference = 21 – 10 = 11 which is divisible by 11.
So, yes, 963391 is divisible by 11.

Method 2: Subtract the last digit from the remaining truncated number. If the result is divisible by 11, then so was the first number. Apply this rule over and over again as necessary.
E.g. 963391 → 96339 – 1 = 96338 → 9633 – 8 = 9625 → 962 – 5 = 957 → 95 – 7 = 88, which is divisible by 11. So yes, 963391 is divisible by 11.

Divisibility Test for 13

To find out if a number is divisible by 13 or not, follow these steps:

1. Separate the last digit from the rest of the number. Let us call the rest of the number the truncated number. The truncated number has one less digit than the original number or the previous truncated number.
2. Multiply the last digit by 4 and add it to the truncated number.
3. Check if this result is sufficiently small so that you can immediately say if this is divisible by 13. If it is divisible by 13, then so was the original number. If it is not divisible by 13, then neither was the original number.
4. If the number is still too large to visually check if it is divisible, apply this rule over and over again as necessary.

E.g. 12675 → 1267 + 20 = 1287 → 128 + 28 = 156 → 15 + 24 = 39, which is divisible by 13. So yes, 12675 is divisible by 13.

Divisibility Test for 17

To find out if a number is divisible by 17 or not, follow these steps:

1. Separate the last digit from the rest of the number. Let us call the rest of the number the truncated number. The truncated number has one less digit than the original number or the previous truncated number.
2. Multiply the last digit by 5 and subtract it from the truncated number.
3. Check if this result is sufficiently small so that you can immediately say if this is divisible by 17. If it is divisible by 17, then so was the original number. If it is not divisible by 17, then neither was the original number.
4. If the number is still too large to visually check if it is divisible, apply this rule over and over again as necessary.

E.g. 21165 → 2116 – 25 = 2091 → 209 – 5 = 204 → 20 – 20 = 0, which is divisible by 17. So yes, 21165 is divisible by 17.

Divisibility Test for 19

Add two times the last digit to the remaining leading truncated number. If the result is divisible by 19, then so was the first number. Apply this rule over and over again as necessary.

E.g. 185117 → 18511 + 2 × 7 = 18525 → 1852 + 2 × 5 = 1862 → 186 + 2 × 2 = 190, which is divisible by 19. So yes, 185117 is divisible by 19.

Divisibility Test for Composites

A number is divisible by a composite if it is also divisible by all the prime factors.
E.g. 157905 is divisible by 33 if it is divisible by 3 AND by 11.

Now, Try It Yourself

Que. 1Which of the following option is completely divisible by 11?

a.107604

b.98637

c.116571

d.80703

e.None of these

Ans. b.98637

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