A number “a” is said to be completely divisible by another number “b” if “a” upon divided by the number “b” leaves no remainder. This means that “a” is divisible by “b” if a ÷ b leaves a remainder 0. Can you determine whether a number is divisible by another number without actually dividing it?

Here we have some rules, also known as divisibility rules that will help you determine the divisibility of a number without actually going through the entire process of division. These rules can also be referred to as shortcuts to check upon the divisibility of a number.

Divisibility by 2

Divisibility Rule – All even numbers are divisible by 2. This means that numbers ending with 0, 2, 4, 6, and 8 are divisible by 2.

For example,

Let us check whether the following numbers are divisible by 2:

202, 305, 638, 7894, 519

Number	Last Digit of the Number	Divisible by 2
202	2	Yes
305	5	No
638	8	Yes
7894	4	Yes
519	9	No

Divisibility by 3

Divisibility Rule – Add the digits of the number whose divisibility by 3 is to be checked. If the sum of these digits is divisible by 3, the number would also be divisible by 3.

For example,

Let us check whether the following numbers are divisible by 3:

345, 796, 6125, 321, 506

Number	Sum of the Digits	Sum Divisible by 3	Number Divisible by 3
345	12	Yes	Yes
796	22	No	No
6125	14	No	No
321	6	Yes	Yes
506	11	No	No

Divisibility by 4

Divisibility Rule – Check the last two digits of the number. If the number formed by these two digits is divisible by 4, or the last two digits are 00, then the complete number would also be divisible by 4.

For example,

Let us check whether the following numbers are divisible by 4:

488, 564, 3128, 6431, 9147

Number	Last Two Digits	Last Two digits Divisible by 4	Number Divisible by 4
488	88	Yes	Yes
500	00	Yes	Yes
3128	28	Yes	Yes
6431	31	No	No
9147	47	No	No

Divisibility by 5

Divisibility Rule – Check the units place of the number. A number that ends in 0 or 5 would be divisible by 5.

For example,

Let us check whether the following numbers are divisible by 5:

185, 269, 7846, 490, 355

Number	Number at unit’s place	Number Divisible by 5
185	5	Yes
269	9	No
7846	6	No
490	0	Yes
355	5	Yes

Divisibility by 6

Divisibility Rule – Check whether the given number is divisible by both 2 and 3. If a number is divisible by both 2 and 3, it is also divisible by 6.

For example,

Let us check whether the following numbers are divisible by 6:

618, 948, 4536, 4172, 6541

Number	Number Divisible by 2	Number divisible by 3	Number Divisible by 6
618	Yes	Yes	Yes
948	Yes	Yes	Yes
4536	Yes	Yes	Yes
4172	Yes	No	No
6541	No	No	No

Divisibility by 7

Divisibility Rule – Check the digit at the unit’s place of the number given. Subtract twice of this digit from the remaining digits. If the number so obtained is divisible by 7, the original number would also be divisible by 7.

For example,

Let us check whether the following numbers are divisible by 7:

749, 357, 5125, 648, 321

Number	Twice of the digit at unit’s place	Number obtained after Subtraction	Resultant Number Divisible by 7	Original Number Divisible by 7
749	9 x 2 = 18	74 – 18 = 56	Yes	Yes
357	7 x 2 = 14	35 – 14 = 21	Yes	Yes
5125	5 x 2 = 10	512 – 10 = 502	No	No
648	8 x 2 = 16	64 – 16 = 48	No	No
321	1 x 2 = 2	32 – 2 = 30	Yes	Yes

Divisibility by 8

Divisibility Rule – Check the last three digits of the number. If the number formed by these three digits is divisible by 8, or the last three digits are 000, then the complete number would also be divisible by 8.

Also, a number that is divisible by both 2 and 4 will be divisible by 8.

For example,

Let us check whether the following numbers are divisible by 8:

161, 25000, 648, 488, 368

Number	Last Three Digits	Last Three digits Divisible by 8	Number Divisible by 8
1610	610	No	No
25000	000	Yes	Yes
20648	648	Yes	Yes
72488	488	Yes	Yes
13368	368	Yes	Yes

Divisibility by 9

Divisibility Rule – Add the digits of the number whose divisibility by 9 is to be checked. If the sum of these digits is divisible by 9, the number would also be divisible by 9.

For example,

Let us check whether the following numbers are divisible by 9:

649, 327, 127, 7120, 3025

Number	Sum of the Digits	Sum Divisible by 9	Number Divisible by 9
649	19	No	No
333	18	Yes	Yes
129	12	No	No
7128	18	Yes	Yes
3025	10	No	No

Divisibility by 10

Divisibility Rule – Check the digit at the unit’s place of the number given. If the number at the unit’s place is 0, the number is divisible by 10.

Let us check whether the following numbers are divisible by 10:

640, 545, 329, 2148, 9410

Number	Number at unit’s place	Number Divisible by 10
640	0	Yes
545	5	No
329	9	No
2148	8	No
9410	0	Yes

It is important to note that apart from the above rules, a number is divisible by a certain number if it is divisible by its factors. For example, a number that is divisible by 2 and 4 would also be divisible by 8. Similarly, a number that is divisible by 2 and 3 will also be divisible by 6.

Here are more examples that will help you clearly visualise the above explanations of the divisibility rules and have a better understanding of these short, yet important topics in mathematics.

Divisibility rules help us work out whether a number is exactly divisible by other numbers (i.e. there is no remainder).

The rules are shortcuts for finding out whether numbers are exactly divisible without doing division calculations. Some of these rules along with examples are illustrated below:

Divisible by 2?

Rule: If it ends with a 0, 2, 4, 6, or 8

Number	Divisible?	Why?
456	Yes	The last digit is 6
68	Yes	The last digit is 8
25	No	The last digit is 5 (not a 2,4,6,or 8)
207	No	The last digit is 7 (not a 2,4,6,or 8)

Divisible by 3?

Rule: If the sum of the digits is a multiple of 3

Number	Divisible?	Why?
405	Yes	4 + 0 + 5 = 9 (9 is a multiple of 3)
381	Yes	3 + 8 + 1 = 12 (12 is a multiple of 3)
928	No	9 + 2 + 8 = 19 (19 is not a multiple of 3)
4,616	No	4 + 6 + 1 + 6 = 17 (17 is not a multiple of 3)

Helper: The multiples of 3 include…

3 6 9 12 15 18 21 24 27 30 33 36 39 42 45

Divisible by 4?

Rule: If the last two digits are a multiple of 4
(or if the last two digits are 00)

Number	Divisible?	Why?
348	Yes	48 is a multiple of 4
27,616	Yes	16 is a multiple of 4
8,514	No	14 is not a multiple of 4
722	No	22 is not a multiple of 4
1,200	Yes	The last two digits are 00 200 is a multiple of 4

Helper: The multiples of 4 include…

4 8 12 16 20 24 28 32 36 40 44 48 52 56 60

Divisible by 5?

Rule: If it ends with a 5 or a 0

Number	Divisible?	Why?
3,425	Yes	The last digit is 5
750	Yes	The last digit is 0
8,551	No	The last digit is 1 (not a 0 or a 5)
394	No	The last digit is 4 (not a 0 or a 5)

Divisible by 6?

Rule: If it is divisible by 2 and by 3

Number	Divisible?	Why?
5,106	Yes	The last digit is a 2 (it is a multiple of 2 ) and…5 + 1 + 0 + 6 = 12 (12 is a multiple of 3)
636	Yes	The last digit is a 6 (it is a multiple of 2 ) and…6 + 3 + 6 = 15 (15 is a multiple of 3)
5,912	No	The last digit is a 2 (it is a multiple of 2 ) but…5 + 9 + 1 + 2 = 17 (17 is not a multiple of 3)
508	No	The last digit is a 8 (it is a multiple of 2 ) but…5 + 0 + 8 = 13 (13 is not a multiple of 3)

Divisible by 9?

Rule: If the sum of the digits are a multiple of 9

Number	Divisible?	Why?
7,686	Yes	7 + 6 + 8 + 6 = 27 (27 is a multiple of 9)
252	Yes	2 + 5 + 2 = 9 (9 is a multiple of 9)
883	No	8 + 8 + 3 = 19 (19 is not a multiple of 9)
5,105	No	5 + 1 + 0 + 5 = 11 (11 is not a multiple of 9)

Helper: The multiples of 9 include…

9 18 27 36 45 54 63 72 81 90 99 108 117 126 135

Divisible by 10?

Rule: If the last digit is 0

Number	Divisible?	Why?
880	Yes	The last digit is 0
9,560	Yes	The last digit is 0
312	No	The last digit is 2 (not a 0)
7,897	No	The last digit is 7 (not a 0)