# Divisibility rules

**2**Divisibility by

**3**Divisibility by

**4**Divisibility by

**5**Divisibility by

**6**Divisibility by

**9**Divisibility by

**10**Divisibility tests

## Divisibility by 2

**The number is divided by 2** when the last digit is even (0, 2, 4, 6, or 8).

2, 8, 16, 24, 66, 150 — divided by **2**, as the last digits of the numbers is even;

3, 7, 19, 35, 77, 453 — not divided by **2**, as the last digit of the number is odd.

## Divisibility by 3

**The number is divided by 3** when the sum of the digits is divided by 3.

75 — divided by **3**, as 7+5=15, and the number 15 is divided by 3 (15:3=5);

471 — divided by **3**, as 4+7+1=12, and the number 12 is divided by 3 (12:3=4);

532 — not divided by **3**, as 5+3+2=10, and the number 10 is not divided by 3 (10:3=3

## Divisibility by 4

**The number is divided by 4** when the number formed by the last two digits is divisible by 4.

**The two-digit number is divisible by 4** when twice the tens digit, plus the ones digit is divisible by 4.

4576 — divided by **4**, as last two digits 76 divided by 4 (7·2+6=20, 20:4=5);

9634 — not divided by **4**, as last two digits 34 not divided by 4 (3·2+4=10, 10:4=2

## Divisibility by 5

**The number is divided by 5** when the last digit is 0 or 5.

375, 5680, 233575 — divided by **5**, as the last digit of this numbers is 0 or 5;

9634, 452, 389753 — not divided by **5**, as the last digit of this numbers isn't 0 or 5.

## Divisibility by 6

**The number is divided by 6** when it divided by 2 and 3.

462 — divided by **6**, as it divided by 2 (last digits of the number is even) and divided by 3 (sum of the digits is divided by 3: 4+6+2=12, 12:3=4);

3456 — divided by **6**, as it divided by 2 (last digits of the number is even) and divided by 3 (sum of the digits is divided by 3: 3+4+5+6=18, 18:3=6);

24642 — divided by **6**, as it divided by 2 (last digits of the number is even) and divided by 3 (sum of the digits is divided by 3: 2+4+6+4+2=18, 18:3=6);

861 — not divided by **6**, as it not divided by 2 (last digits of the number isn't even);

3458 — not divided by **6**, as it divided by 3 (sum of the digits is not divided by 3: 3+4+5+8=20, 20:3=6

34681 — not divided by **6**, as it divided by 2 (last digits of the number isn't even).

## Divisibility by 9

**The number is divided by 9** when the sum of the digits is divided by 9.

69759 — divided by **9**, as sum of the digits is divided by 9 (6+9+7+5+9=36, 36:9=4);

34681 — not divided by **9**, as sum of the digits is not divided by 9 (3+4+6+8+1=22, 22:9=2

## Divisibility by 10

**The number is divided by 10** when the last digit is 0.

460, 24000, 1245464570 — divided by **10**, as the last digit of this numbers is 0;

234, 25048, 1230000003 — не делятся на **10**, as the last digit of this numbers isn't 0.

## Divisibility by 11

**The number is divided by 11** if the sum of digits standing in even places is equal to the sum of digits standing in odd places or divided by 11.

242 — is divided by **11**, as the sum of the numbers in odd positions S_{2n+1} = 2 + 2 = 4; sum of digits in even positions S_{2n} = 4 and S_{2n+1} = S_{2n}.

319 — is divided by **11**, as the sum of the numbers in odd positions S_{2n+1} = 3 + 9 = 12; sum of digits in even positions S_{2n} = 1, and their difference S_{2n+1} - S_{2n} = 11 - divided by 11.

919380 — is divided by **11**, as the sum of the numbers in odd positions S_{2n+1} = 9 + 9 + 8 = 26; sum of digits in even positions S_{2n} = 1 + 3 + 0 = 4, and their difference S_{2n+1} - S_{2n} = 22 - divided by 11.

2838 — is divided by **11**, as the sum of the numbers in odd positions S_{2n+1} = 2 + 3 = 5; sum of digits in even positions S_{2n} = 8+ 8 = 16, and their difference S_{2n} - S_{2n+1} = 11 - divided by 11.

244 — is not divided by **11**, as the sum of the numbers in odd positions S_{2n+1} = 2 + 4 = 6; sum of digits in even positions S_{2n} = 4 and their difference S_{2n+1} - S_{2n} = 2 - not divided by 11.

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