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Divisibility Rules

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

NumberLast Digit of the NumberDivisible by 2
2022Yes
3055No
6388Yes
78944Yes
5199No

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

NumberSum of the DigitsSum Divisible by 3Number Divisible by 3
34512YesYes
79622NoNo
612514NoNo
3216YesYes
50611NoNo

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

NumberLast Two DigitsLast Two digits Divisible by 4Number Divisible by 4
48888YesYes
50000YesYes
312828YesYes
643131NoNo
914747NoNo

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

NumberNumber at unit’s placeNumber Divisible by 5
1855Yes
2699No
78466No
4900Yes
3555Yes

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

NumberNumber Divisible by 2Number divisible by 3Number Divisible by 6
618YesYesYes
948YesYesYes
4536YesYesYes
4172YesNoNo
6541NoNoNo

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

NumberTwice of the digit at unit’s placeNumber obtained after SubtractionResultant Number Divisible by 7Original Number Divisible by 7
7499 x 2 = 1874 – 18 = 56YesYes
3577 x 2 = 1435 – 14 = 21YesYes
51255 x 2 = 10512 – 10 = 502NoNo
6488 x 2 = 1664 – 16 = 48NoNo
3211 x 2 = 232 – 2 = 30YesYes

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

NumberLast Three DigitsLast Three digits Divisible by 8Number Divisible by 8
1610610NoNo
25000000YesYes
20648648YesYes
72488488YesYes
13368368YesYes

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

NumberSum of the DigitsSum Divisible by 9Number Divisible by 9
64919NoNo
33318YesYes
12912NoNo
712818YesYes
302510NoNo

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

NumberNumber at unit’s placeNumber Divisible by 10
6400Yes
5455No
3299No
21488No
94100Yes

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)

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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

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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

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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)

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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)

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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

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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)

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