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Unit: Number System
Chapter: Divisibility of Numbers
Reference: – Divisibility Tests for 2, 3, 4, 5, 6, 8, 9, 10, and 11, Divisibility by 7 and 13 (Advanced), Place Value Proofs, Application in Factorisation and Simplification
After studying this chapter, you should be able to understand:
- Divisibility Test for 2 (Even Numbers)
- Divisibility Test for 3 and 9 (Digit Sum)
- Divisibility Test for 4 and 8 (Last Digits)
- Divisibility Test for 5 and 10 (Last Digit)
- Divisibility Test for 6 (Combination of 2 and 3)
- Divisibility Test for 11 (Alternating Digit Sum)
- Advanced Tests: Divisibility by 7 and 13
- Why Divisibility Rules Work (Place Value Explanation)
Tests of Divisibility
Divisibility tests are used to find whether one number is divisible by another without carrying any actual division. Let us look at the basic tests of divisibility.
- Basic Tests of Divisibility
- Divisibility by 2– For a number to be divisible by 2, the digit at the unit place should be 0, 2, 4, 6 or 8 i.e. the number should be even.
- Divisibility by 3– A number is divisible by 3, if the sum of all of its digits is divisible by 3.
- Divisibility by 4- A number is divisible by 4, if the last two digits of that number is divisible by 4.
- Divisibility by 5- A number is divisible by 5, if the last digit of the number is either 0 or 5.
- Divisibility by 6- A number is divisible by 6, if the number is divisible by both 2 & 3.
- Divisibility by 7- For a number is divisible by 7, first we take the last digit of the number and multiply it by 2 and then we subtract this number from the remaining digits of the number. If the subtracted number is 0 or is divisible by 7, then the actual number is also divisible by 7.
- Divisibility by 8- A number is divisible by 8, if the last three digits of the number is divisible by 8.
- Divisibility by 9- A number is divisible by 9, if the sum of its digit is divisible by 9.
- Divisibility by 10- A number is divisible by 10, if the last digit of the number is 0.
- Divisibility by 11- For a number is divisible by 11, first we sum up the digits of the number at odd places and at even places respectively, then we find the difference of both. If the difference is either 0 or divisible by 11, then the number is divisible by 11.
- Divisibility by 12- A number is divisible by 12, if the number is divisible by both 3 & 4.
- General Rules of Divisibility
- If a number is divisible by another number, then it is also divisible by all the factors of the other number.
Example- 30 is divisible by 6.
Factors of 6 are 2 and 3.
So, 30 is also divisible by 2 and 3.
- If two numbers are divisible by another number, then their sum or the difference will also be divisible by the other number.
Example- 28 and 12 are divisible by 4.
Sum = 28 + 12 = 40 which is also divisible by 4.
Difference = 28 – 12 = 16 which is also divisible by 4.
- If a number is divisible by two co-prime numbers, then it is also divisible by the product of the two co-prime numbers.
Example- 360 is divisible by both the co=primes 9 and 4.
Product of co-primes = 9 × 4 = 36.
So, 360 is also divisible by 36.
