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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=12, and the number 12 is divided by 3 (12:3=4);

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.

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