RULES OF DIVISIBILITY
A divisibility rule is a shorthand and useful way of determining whether a given
integer is divisible by a fixed divisor without performing the division, usually by
examining its digits.
Divisibility Rule of 1
Every number ever is divisible by 1.
Divisibility Rule of 2
Every even number is divisible by 2. Any number that ends with 2, 4, 6, 8, or 0 gives 0
as the remainder when divided by 2.
For example, 12, 46, and 780 are all divisible by 2.
Divisibility Rules of 3
A number is completely divisible by 3 if the sum of its digits is divisible by 3. You can
also repeat this rule until you get a single-digit sum.
Divisibility Rule of 4
If the number formed by the last two digits of a number is divisible by 4, then that
number is divisible by 4. Numbers having 00 as their last digits are also divisible by 4.
Example 1: Consider the number 284. Check the last two digits.
The last two digits of the number form the number 84. As 84 is divisible by 4, the
original number 284 is also divisible by 4.
Divisibility Rule of 5
If a number ends with 0 or 5, it is divisible by 5.
Divisibility Rule of 6
If a number is divisible by 2 and 3, it will be divisible by 6 as well.
Divisibility Rules of 7
If subtracting twice of the last digit from the number formed by remaining digits is 0
or divisible by 7, the number is divisible by 7.
Example: Check whether 905 is divisible by 7 or not.
Step 1: Check the last digit and double it.
Last digit =5
Multiply it by 2.
5×2=10
Step 2: Subtract this product from the rest of the number.
Here, the remaining number =90
90−10=80
Step 3: If this number is 0 or a multiple of 7, then the original number is also divisible
by 7.
80 is not divisible by 7. So, 905 is also not divisible by 7.
Divisibility Rule of 8
If the number formed by the last three digits of a number is divisible by 8, we say
that the number is divisible by 8.
Example 1: In the number 4176, the last 3 digits are 176.
If we divide 176 by 8, we get:
Since 176 is divisible by 8, 4176 is also divisible by 8.
Example 2:
Thus, 12,920 is divisible by 8.
Divisibility Rule of 9
If the sum of digits of the number is divisible by 9, then the number itself is divisible
by 9. You can keep adding further by repeating the rule. If the single-digit sum is 9,
the number is divisible by 9.
Divisibility Rule of 10
Any number whose last digit is 0 is divisible by 10.
Divisibility Rule for 11
If the difference of the sum of alternative digits of a number is divisible by 11, then
that number is divisible by 11.
Example 1: Consider the number. 2846767. First, understand the digit positions. We
find two sums: the sum of digits at the even places and the sum of digits at the odd
places.
Sum of digits at even places (From right) =8+6+6=20
Sum of digits at odd places (From right) =7+7+4+2=20
Difference =20–20=0
Difference is divisible by 11.
Thus, 2846767 is divisible by 11.
Example 2: Is 61809 divisible by 11?
Identify digits in odd places and digits in even places.
Here, 6+8+9=23 and 0+1=1
Difference =23–1=22
22 is divisible by 11.
Thus, the given number is divisible by 11.
Another Divisibility Rule For 11
There’s another simple divisibility rule for 11.
Subtract the last digits from the remaining number. Keep doing this until we get a twodigit number. If the number obtained is divisible by 11, the original number is divisible
by 11.
Example: 1749
174−9=165
16−5=11 … divisible by 11
Thus, 1749 is divisible by 11.