Divisibility Rules

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1.6Divisibility Rules


In these revision notes for Divisibility Rules, we cover the following key points:

  • What does divisibility mean?
  • How do we write in symbols when a number is divisible by another number?
  • What are the divisibility rules for the numbers from 1 to 20?
  • What are relatively prime numbers?
  • How do relatively prime numbers determine some of divisibility rules?

Divisibility Rules Revision Notes

An integer x is divisible by another integer y if the result of x ÷ y is another integer, i.e. it is a number without remainder (r = 0). We write the symbol (⁝) to represent the divisibility of two numbers.

The divisibility by 1 rule states that all numbers are divisible by 1.

The divisibility by 2 rule states that the number must be even in order to be divisible by 2.

The divisibility by 3 rule states that the sum of digits of the original number must be divisible by 3.

The divisibility by 4 rule states that the last two digits of the original number must form a number divisible by 4.

The divisibility by 5 rule states that a number must end with 0 or 5.

The divisibility by 6 rule states that a number must be divisible by 2 and 3 at the same time in order to be divisible by 6.

The divisibility by 7 rule states that a number is divisible by 7 if the difference between twice the value of the digit in the ones place and the number formed by the rest of the digits is either 0 or a multiple of 7.

The divisibility by 8 rule states that the last three digits of the original number must form a number that is divisible by 8.

The divisibility by 9 rule states that the sum of digits of the original number must be divisible by 9.

The divisibility by 10 rule states that the number must end with zero to be divisible by 10.

The divisibility by 11 rule states that a number is divisible by 11 if the difference between the sum of the digits in the odd place value and even place value is a multiple of 11 (including 0).

The divisibility by 12 rule states that a number must be divisible by both 3 and 4 to be divisible by 12.

The divisibility by 13 rule states that a number is divisible by 13 if after grouping the digits in groups of three starting from the rightmost place value and applying the subtraction and addition of the numbers obtained by these groups alternatively from right to left, we obtain a number divisible by 13, including 0.

The divisibility by 14 rule states that a number is divisible by 14 if it is divisible by both 2 and 7.

The divisibility by 15 rule states that a number is divisible by 15 if it is divisible by both 3 and 5.

The divisibility by 16 rule states that a number is divisible by 16 if the last three digits form a number that is divisible by 16 while the fourth last digit is even.

The divisibility by 17 rule states that a number is divisible by 17 if after multiplying the last digit by 5 and subtract it from the rest, the result is divisible by 17.

The divisibility by 18 rule states that a number is divisible by 18 if it is divisible by both 2 and 9.

The divisibility by 19 rule states that a number is divisible by 19 if twice the last digit plus the rest of number give a number divisible by 19.

The divisibility by 20 rule states that a number is divisible by 20 if the last two digits of the original number are a multiple of 20.

Two numbers are relatively prime when they don't have any other common factor besides 1. In general, if a number is divisible by each of two relatively prime numbers, it is also divisible by their product.

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