Divisibility rules are a set of guidelines or criteria that can be used to determine whether a given number is divisible by another number without having to perform the actual division. These rules are based on properties of numbers, such as their digits or the sum of their digits and can be used as a quick way to check for divisibility without having to perform the calculation.
In this article, we will learn about divisibility rules from 2 to 13.
Some of us consider math to be difficult. Sometimes it's necessary to use tricks and shortcuts to solve math problems quickly and simply without lengthy calculations. These rules are a good example of such shorthand techniques.
Let's discuss the math divisibility rules with examples.
Divisibility rule for 2 states that a number is divisible by 2, if its unit digit (the digit in the ones place) is an even number 2, 4, 6, 8 including 0.
Example:
a) The number 42 is divisible by 2 because its unit digit is 2, which is an even number.
b) The number 71 is not divisible by 2 because its unit digit is 1, which is not an even number.
A divisibility rule for 3 states that if the sum of the digits of a number is divisible by 3, then the whole number is divisible by 3.
Example:
a) The number 153 is divisible by 3 because the sum of the digits (1 + 5 + 3) is 9, which is divisible by 3.
b) The number 622 is not divisible by 3 because the sum of the digits (6 + 2 + 2) is 10, which is not divisible by 3.
c) The number 729 is divisible by 3 because the sum of the digits (7 + 2 + 9) is 18, which is divisible by 3.
The divisibility rule for 4 states that if the last two digits of a number are divisible by 4, then the whole number is completely divisible by 4.
Example:
a) The number 1936 is divisible by 4 because the last two digits, "36" are also divisible by 4.
b) The number 1248 is divisible by 4 because the last two digits, "48" are also divisible by 4.
c) The number 734 is not divisible by 4 because the last two digits, "34" are not divisible by 4.
The divisibility rule for 5 states that if the unit digit (the digit in the ones place) of a number is either 0 or 5, then the whole number is divisible by 5.
Example: Is the number 535 divisible by 5?
-> To check, we look at the unit digit, which is 5.
-> Since 5 is either 0 or 5, we know that 535 is divisible by 5.
The divisibility rule for 6 states that if the number is divisible by both 2 and 3, then the whole number is divisible by 6.
Example: Is 1542 divisible by 6?
-> To check, we first need to see if it is divisible by 2 (a multiple of 2) and 3 (a multiple of 3).
->1542 is an even number, so it is divisible by 2.
-> To check if it is divisible by 3, we need to add up the digits (1 + 5 + 4 + 2 = 12) and see if the sum is divisible by 3. Since 12 is divisible by 3, 1542 is also divisible by 3.
-> Since 1542 is divisible by both 2 and 3, it is also divisible by 6.
Take the last digit of the number and double it. Subtract this doubled digit from the remaining part of the number (i.e., the number without its last digit). If the result is 0 and the multiple of 7 then the original number is also divisible by 7. If not, the original number is not divisible by 7.
Example: Let's check if the number 728 is divisible by 7.
-> We would first take unit digit 8 and double it, which is 8 x 2 = 16.
-> Then, we would subtract that from the remaining digits, 72 - 16 = 56.
-> Since 56 is divisible by 7, it means the whole number 728 is also divisible by 7.
The divisibility rule for 8 states that if the last three digits of a number are divisible by 8, then the entire number is divisible by 8.
Example: Is the number 12848 divisible by 8?
-> To check, we look at the last three digits (848) and see it is divisible by 8.
-> So 12848 is also divisible by 8.
The divisibility rule for 9 states that if the sum of its digits of a number is divisible by 9, then the whole number is divisible by 9.
Example:
-> Let's take the number 153. The sum of its digits is 1 + 5 + 3 = 9.
-> Since 9 is divisible by 9,153 is also divisible by 9.
The divisibility rule for 10 states that the unit digit is 0 of a number, then the number is divisible by 10.
Example:
a) Is the number 60 divisible by 10?
-> Yes, because its unit digit is 0.
b) Is the number 57 divisible by 10?
-> No, because its unit digit is not 0.
The rule states that if the difference between the sum of the digits in the odd positions and the sum of the digits in the even positions of a number is divisible by 11 or 0, then the original number is also divisible by 11.
Example: Let's take the number 9328.
-> The sum of the digits in the odd positions (9 + 2) is 11.
-> The sum of the digits in the even positions (3 + 8) is 11.
-> The difference between these two sums (11 - 11) is 0.
-> Since the difference is 0, this means 9328 is divisible by 11.
Let's take the number 74668.
-> The sum of the digits in the odd positions (7 + 6 + 8) is 21.
-> The sum of the digits in the even positions (4 + 6) is 10.
-> The difference between these two sums (21 - 10) is 11.
-> Since the difference is 11, this means 9328 is divisible by 11.
The divisibility rule by 12 states that a number must satisfy both the conditions of being divisible by 3 and divisible by 4 to be considered divisible by 12.
Example: Is 948 divisible by 12?
-> To check if 948 is divisible by 12, we first check if it is divisible by 3 and 4.
-> 948 is divisible by 3 (Since 9 + 4 + 8 = 21 which is divisible by 3, this means 948 is divisible by 3.)
-> 948 is divisible by 4 (Since the last two digits 48 is divisible by 4, this means 948 is divisible by 4.)
-> Since 948 is divisible by both 3 and 4, it is also divisible by 12.
-> Therefore, 948 ÷ 12 = 79.
The divisibility rule for 13 can be easily understood by the following steps.
To determine whether a number is divisible by 13, follow these steps:
-> Take the last digit of the number and multiply it by 4.
-> Add this result to the remaining digits of the number.
-> Repeat steps 1 and 2 with the new number obtained in step 2 until you have a two-digit number.
-> Check if the two-digit number is divisible by 13. If it is, then the original number is divisible by 13.
Let's check if the number 5512 is divisible by 13.
-> First, take unit digit and multiplied by 4, which is 2 x 4 = 8.
-> Add the resultant to the remaining digits, 551 + 8 = 559.
-> Now, repeat this again and take unit digit 9 and multiplied by 4, which is 9 x 4 = 36.
-> Add the resultant to the remaining digits, 55 + 36 = 91.
Since 91 is divisible by 13, it means the whole number 5512 is also divisible by 13.
In this chart, you can see all divisibility tests in a glance.
Divisibility Rules | |
---|---|
A number is divisble by | |
2 | If last digit is 0, 2, 4, 6, or 8 |
3 | If the sum of the digits is divisible by 3 |
4 | If the last two digits is divisible by 4 |
5 | If the last digit is 0 or 5 |
6 | If the number is divisible by 2 and 3 |
7 | cross off last digit, double it and subtract. Repeat if you want. If new number is divisible by 7, the original number is divisible by 7 |
8 | If last 3 digits is divisible by 8 |
9 | If the sum of the digits is divisible by 9 |
10 | If the last digit is 0 |
11 | Subtract the last digit from the number formed by the remaining digits. If new number is divisible by 11, the original number is divisible by 11 |
12 | If the number is divisible by 3 and 4 |
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