Divisibility Rules – Number System

What is Divisibility Rule?

A Divisibility Rule or Test is the way of determining or testing whether a particular number is divisible by another number without performing the division. Each fixed number has its own divisibility rule that makes us clear about a particular number that it is divisible by a number or not.

Following are the Divisibility Rules for various numbers:

Divisibility Rule for 2:

Any number is divisible by 2 if it ends with even numbers i.e., 0,2,4,6,8.

Divisibility Rule for 4:

Any number is divisible by 4 if the number formed by the last two digits of a number is exactly divisible by 4.

Example: 216 is divisible by 4 as the number formed by the last two digits i.e.,16 is exactly divisible by 4.

Divisibility Rule for 8:

Any number is divisible by 8 if the number formed by the last three digits of a number is exactly divisible by 8.

Example: 6,120 is divisible by 8 as the number formed by the last three digits i.e., 120 is exactly divisible by 8.

Number system concept for Divisibility rules and divisibility test for 2,4, and 8.

Divisibility Rule for 3:

Any number is divisible by 3 if the sum of the digits of a number is divisible by 3.

Example: 156 is divisible by 3 since the sum of the digits is 12 (1+5+6=12), and 12 is divisible by 3.

Divisibility Rule for 9:

Any number is divisible by 9 if the sum of the digits of a number is divisible by 9.

Example: 549 is divisible by 9 since the sum of the digits is 18 (5+4+9=18), and 18 is divisible by 9.

Number system concept for Divisibility rules and divisibility test for 3, 9, and 99.

Divisibility Rule for 5:

Any number is divisible by 5 if the last digit of the number is either 0 or 5.

Example: 225 is divisible by 5 since the last digit is 5.

Divisibility Rule for 10:

Any number is divisible by 10 if the last digit of a number is 0.

Example: 1470 is divisible by 10 since the last digit is 0.

Divisibility Rule for 25:

Any number is divisible by 25, the number formed by the last two digits of the number must be 00, 25, 50, or 75 (basically, last two digits should be divisible by 25).

Example: 1,234,175 is divisible by 25 because the last two digits form 75.

Number system concept for Divisibility rules and divisibility tests for 5 and 25

Divisibility Rule for 6:

Any number is divisible by 6 when it is divisible by both 2 AND 3 at the same time.

Example: 168 is divisible by 6 since it is divisible by 2 AND it is divisible by 3.

Now answer the following question based on the above logic:

Divisibility Rule and divisibility test for 6
Divisibility Rule and divisibility test for 6

Divisibility rule for 11:

Take the alternating sum of the digits in the number, read from left to right. If that is divisible by 11, so is the original number.

For example, 3927 has alternating sum of digits 3-9+2-7 = -11. Since -11 is divisible by 11, so is 3927.

Similarly, for 3415, the alternating sum of digits is 3-4+1-5 = -5. This is not divisible by 11, so neither is 3415.

Divisibility Rule for 12:

This will be similar to the rule for 6. Any number which is divisible by both 3 AND 4 is divisible by 12. That means, the sum of its digits should be divisible by 3 and its last two digits should be divisible by 4.

In this way, we can calculate whether the given number is divisible by 2, 4, 5, 6, 8, 9, 10, 11, 12, 15, 25, 99 …etc, without doing the actual division. I hope this article helped in explaining the divisibility rule concept.

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