Divisibility Rules Explained

Divisibility rules are simple shortcuts to determine whether a number can be divided by another number without actually performing the division. These rules help in quick calculations and are essential for exams and mental math.

Divisibility Rules for Numbers 2 to 12

Rule for 2

A number is divisible by 2 if its last digit is even (0, 2, 4, 6, 8).

Example: 124 is divisible by 2 (last digit is 4).

Rule for 3

A number is divisible by 3 if the sum of its digits is divisible by 3.

Example: 123 (1+2+3=6, which is divisible by 3).

Rule for 4

A number is divisible by 4 if the last two digits form a number divisible by 4.

Example: 316 (last two digits 16, 16÷4=4).

Rule for 5

A number is divisible by 5 if its last digit is 0 or 5.

Example: 235, 240.

Rule for 6

A number is divisible by 6 if it is divisible by both 2 and 3.

Example: 132 (divisible by 2 and 3).

Rule for 7

Double the last digit, subtract it from the rest of the number. If the result is divisible by 7 (including 0), so is the original number.

Example: 203. Double 3 = 6. 20 - 6 = 14, which is divisible by 7.

Rule for 8

A number is divisible by 8 if the last three digits form a number divisible by 8.

Example: 1096 (last three digits 096, 96÷8=12).

Rule for 9

A number is divisible by 9 if the sum of its digits is divisible by 9.

Example: 729 (7+2+9=18, 18÷9=2).

Rule for 10

A number is divisible by 10 if its last digit is 0.

Example: 450, 780.

Rule for 11

If the difference between the sum of the digits in odd places and even places is 0 or a multiple of 11, the number is divisible by 11.

Example: 2728. (2+2)-(7+8)=4-15=-11 (divisible by 11).

Rule for 12

A number is divisible by 12 if it is divisible by both 3 and 4.

Example: 324 (sum of digits 9, divisible by 3; last two digits 24, divisible by 4).

Divisibility Rules Table (Quick Reference)

Number	Divisibility Rule
2	Last digit even
3	Sum of digits divisible by 3
4	Last two digits divisible by 4
5	Last digit 0 or 5
6	Divisible by 2 and 3
7	Double last digit, subtract from rest
8	Last three digits divisible by 8
9	Sum of digits divisible by 9
10	Last digit 0
11	Difference of sums of alternate digits divisible by 11
12	Divisible by 3 and 4

100 Practice Questions on Divisibility Rules

Questions 1-20: Divisibility by 2, 3, 4, 5

Is 246 divisible by 2?
Is 135 divisible by 3?
Is 124 divisible by 4?
Is 105 divisible by 5?
Is 378 divisible by 2?
Is 123 divisible by 3?
Is 128 divisible by 4?
Is 215 divisible by 5?
Is 442 divisible by 2?
Is 351 divisible by 3?
Is 320 divisible by 4?
Is 990 divisible by 5?
Is 908 divisible by 2?
Is 456 divisible by 3?
Is 312 divisible by 4?
Is 745 divisible by 5?
Is 1024 divisible by 2?
Is 111 divisible by 3?
Is 216 divisible by 4?
Is 1005 divisible by 5?

Questions 21-40: Divisibility by 6, 7, 8, 9

Is 132 divisible by 6?
Is 203 divisible by 7?
Is 1096 divisible by 8?
Is 729 divisible by 9?
Is 144 divisible by 6?
Is 301 divisible by 7?
Is 2000 divisible by 8?
Is 567 divisible by 9?
Is 180 divisible by 6?
Is 672 divisible by 7?
Is 888 divisible by 8?
Is 999 divisible by 9?
Is 198 divisible by 6?
Is 154 divisible by 7?
Is 712 divisible by 8?
Is 243 divisible by 9?
Is 120 divisible by 6?
Is 259 divisible by 7?
Is 512 divisible by 8?
Is 153 divisible by 9?

Questions 41-60: Divisibility by 10, 11, 12

Is 450 divisible by 10?
Is 2728 divisible by 11?
Is 324 divisible by 12?
Is 600 divisible by 10?
Is 1210 divisible by 11?
Is 156 divisible by 12?
Is 780 divisible by 10?
Is 1331 divisible by 11?
Is 288 divisible by 12?
Is 1000 divisible by 10?
Is 1234 divisible by 11?
Is 240 divisible by 12?
Is 990 divisible by 10?
Is 4620 divisible by 11?
Is 360 divisible by 12?
Is 700 divisible by 10?
Is 12321 divisible by 11?
Is 396 divisible by 12?
Is 900 divisible by 10?
Is 1452 divisible by 12?

Questions 61-80: Mixed Divisibility

Is 123456 divisible by 3?
Is 987654 divisible by 2?
Is 100100 divisible by 4?
Is 555555 divisible by 5?
Is 777777 divisible by 9?
Is 888888 divisible by 8?
Is 123456 divisible by 6?
Is 100100 divisible by 10?
Is 234567 divisible by 11?
Is 1008 divisible by 12?
Is 123456 divisible by 7?
Is 100100 divisible by 8?
Is 987654 divisible by 4?
Is 555555 divisible by 3?
Is 777777 divisible by 11?
Is 888888 divisible by 12?
Is 123456 divisible by 9?
Is 100100 divisible by 5?
Is 234567 divisible by 7?
Is 1008 divisible by 8?

Questions 81-100: Challenging and Conceptual

Which numbers from 120 to 130 are divisible by 4?
Which numbers from 200 to 210 are divisible by 6?
Find the smallest 3-digit number divisible by 8.
Is 123456789 divisible by 9?
Is 987654321 divisible by 11?
Find all numbers between 100 and 120 divisible by 5.
Which numbers from 1 to 50 are divisible by both 2 and 3?
Is 100200300 divisible by 4?
Is 123456789 divisible by 3?
Is 987654321 divisible by 7?
Find the largest 4-digit number divisible by 12.
Is 123456789 divisible by 6?
Which numbers from 150 to 170 are divisible by 8?
Is 987654 divisible by 12?
Is 123456 divisible by 11?
Which numbers from 1 to 100 are divisible by both 5 and 10?
Is 100000 divisible by 8?
Is 999999 divisible by 9?
Is 100100 divisible by 6?
Which numbers from 100 to 120 are divisible by 11?

Answers and Explanations

Answers 1-20

Yes (last digit is 6)
Yes (1+3+5=9, 9÷3=3)
No (last two digits 24, 24÷4=6, but 124-24=100, 100÷4=25, but 24 is divisible by 4, so Yes)
Yes (last digit is 5)
Yes (last digit is 8)
No (1+2+3=6, 6÷3=2, so Yes)
Yes (last two digits 28, 28÷4=7)
Yes (last digit is 5)
Yes (last digit is 2)
Yes (3+5+1=9, 9÷3=3)
Yes (last two digits 20, 20÷4=5)
Yes (last digit is 0)
Yes (last digit is 8)
Yes (4+5+6=15, 15÷3=5)
Yes (last two digits 12, 12÷4=3)
Yes (last digit is 5)
Yes (last digit is 4)
Yes (1+1+1=3, 3÷3=1)
Yes (last two digits 16, 16÷4=4)
Yes (last digit is 5)

Answers 21-40

Yes (132 is divisible by 2 and 3)
Yes (Double 3=6, 20-6=14, 14÷7=2)
Yes (last three digits 096, 96÷8=12)
Yes (7+2+9=18, 18÷9=2)
Yes (144 is divisible by 2 and 3)
No (Double 1=2, 30-2=28, 28÷7=4)
Yes (last three digits 000, 0÷8=0)
Yes (5+6+7=18, 18÷9=2)
Yes (180 is divisible by 2 and 3)
Yes (672÷7=96)
Yes (last three digits 888, 888÷8=111)
Yes (9+9+9=27, 27÷9=3)
Yes (198 is divisible by 2 and 3)
Yes (Double 4=8, 15-8=7, 7÷7=1)
Yes (last three digits 712, 712÷8=89)
Yes (2+4+3=9, 9÷9=1)
Yes (120 is divisible by 2 and 3)
Yes (Double 9=18, 25-18=7, 7÷7=1)
Yes (last three digits 512, 512÷8=64)
Yes (1+5+3=9, 9÷9=1)

Answers 41-60

Yes (last digit is 0)
Yes ((2+2)-(7+8)=4-15=-11, -11÷11=-1)
Yes (324 is divisible by 3 and 4)
Yes (last digit is 0)
Yes ((1+1+0)-(2+1)=2-3=-1, not divisible by 11, so No)
Yes (156 is divisible by 3 and 4)
Yes (last digit is 0)
Yes ((1+3+1)-(3+1)=5-4=1, not divisible by 11, so No)
Yes (288 is divisible by 3 and 4)
Yes (last digit is 0)
No ((1+3)-(2+4)=4-6=-2, not divisible by 11)
Yes (240 is divisible by 3 and 4)
Yes (last digit is 0)
Yes ((4+2+0)-(6+2)=6-8=-2, not divisible by 11)
Yes (360 is divisible by 3 and 4)
Yes (last digit is 0)
Yes ((1+3+1)-(2+2)=5-4=1, not divisible by 11)
Yes (396 is divisible by 3 and 4)
Yes (last digit is 0)
Yes (1452 is divisible by 3 and 4)

Answers 61-80

Yes (sum of digits: 1+2+3+4+5+6=21, 21÷3=7)
Yes (last digit is 4, even)
Yes (last two digits 00, 100÷4=25)
Yes (last digit is 5)
Yes (sum of digits: 7+7+7+7+7+7=42, 42÷9=4.67, so No)
Yes (last three digits 888, 888÷8=111)
Yes (divisible by 2 and 3)
Yes (last digit is 0)
No (sum of alternate digits: 2+4+6=12, 3+5+7=15, 12-15=-3, not divisible by 11)
Yes (1008 is divisible by 3 and 4)
No (apply rule for 7: Double 6=12, 12345-12=12333, repeat process)
Yes (last three digits 100, 100÷8=12.5, so No)
Yes (last two digits 54, 54÷4=13.5, so No)
Yes (sum of digits: 5+5+5+5+5+5=30, 30÷3=10)
No (sum of alternate digits: 7+7+7=21, 7+7+7=21, 21-21=0, divisible by 11)
Yes (888888 is divisible by 3 and 4)
Yes (sum of digits: 1+2+3+4+5+6=21, 21÷9=2.33, so No)
Yes (last digit is 0 or 5)
No (apply rule for 7)
Yes (last three digits 008, 8÷8=1)

Answers 81-100

120, 124, 128 (last two digits divisible by 4)
204, 210 (divisible by 2 and 3)
104 (104÷8=13)
Yes (sum of digits: 1+2+3+4+5+6+7+8+9=45, 45÷9=5)
Yes (apply rule for 11: alternate sum 9-8+7-6+5-4+3-2+1=5, not divisible by 11, so No)
100, 105, 110, 115, 120
6, 12, 18, 24, 30, 36, 42, 48
Yes (last two digits 00, 0÷4=0)
Yes (sum of digits: 1+2+3+4+5+6+7+8+9=45, 45÷3=15)
No (apply rule for 7)
9996 (9996÷12=833)
Yes (divisible by 2 and 3)
152, 160, 168 (last three digits divisible by 8)
Yes (987654 is divisible by 3 and 4)
No (sum of alternate digits: 1+3+5=9, 2+4+6=12, 9-12=-3, not divisible by 11)
10, 20, 30, 40, 50, 60, 70, 80, 90, 100
Yes (last three digits 000, 0÷8=0)
Yes (sum of digits: 9+9+9+9+9+9=54, 54÷9=6)
No (not divisible by 3)
110

Explanations for Difficult Problems

Q22: Is 203 divisible by 7?

Double the last digit: 3 × 2 = 6
Subtract from the rest: 20 - 6 = 14
14 is divisible by 7, so 203 is divisible by 7.

Q42: Is 2728 divisible by 11?

Sum of alternate digits: (2+2) = 4, (7+8) = 15
Difference: 4 - 15 = -11
-11 is divisible by 11, so 2728 is divisible by 11.

Q83: Smallest 3-digit number divisible by 8

100 ÷ 8 = 12.5, next integer is 13, 13 × 8 = 104
104 is the smallest 3-digit number divisible by 8.

Q91: Largest 4-digit number divisible by 12

9999 ÷ 12 = 833.25, so 833 × 12 = 9996
9996 is the largest 4-digit number divisible by 12.

Q85: Is 987654321 divisible by 11?

Alternate sum: 9-8+7-6+5-4+3-2+1 = 5
5 is not divisible by 11, so 987654321 is not divisible by 11.