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Divisibility rules

A divisibility rule
is a algorithm of determining whether a given number is divisible by a fixed divisor without performing the division.

Divisibility by 2

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

Examples:

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.

Examples:

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=313).

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.

Examples:

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=212).

Divisibility by 5

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

Examples:

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.

Examples:

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=623);

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.

Examples:

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=249).

Divisibility by 10

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

Examples:

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.

Examples:

242 — is divided by 11, as the sum of the numbers in odd positions S2n+1 = 2 + 2 = 4; sum of digits in even positions S2n = 4 and S2n+1 = S2n.

319 — is divided by 11, as the sum of the numbers in odd positions S2n+1 = 3 + 9 = 12; sum of digits in even positions S2n = 1, and their difference S2n+1 - S2n = 11 - divided by 11.

919380 — is divided by 11, as the sum of the numbers in odd positions S2n+1 = 9 + 9  + 8 = 26; sum of digits in even positions S2n = 1 + 3 + 0 = 4, and their difference S2n+1 - S2n = 22 - divided by 11.

2838 — is divided by 11, as the sum of the numbers in odd positions S2n+1 = 2 + 3 = 5; sum of digits in even positions S2n = 8+ 8 = 16, and their difference S2n - S2n+1 = 11 - divided by 11.

244 — is not divided by 11, as the sum of the numbers in odd positions S2n+1 = 2 + 4 = 6; sum of digits in even positions S2n = 4 and their difference S2n+1 - S2n = 2 - not divided by 11.

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