विभाज्यता के नियम
Divisibility Rules
Divisibility rules let a child decide whether one number divides another without doing the division. In an examination with no calculator and little time, they are the fastest tool available for simplifying fractions, finding factors and checking answers.
The rules you must know
| Divisible by | Rule | Example |
|---|---|---|
| 2 | Last digit is 0, 2, 4, 6 or 8 | 1,346 |
| 3 | Sum of digits is divisible by 3 | 471 (4+7+1 = 12) |
| 4 | Last two digits form a number divisible by 4 | 3,916 (16) |
| 5 | Last digit is 0 or 5 | 2,485 |
| 6 | Divisible by both 2 and 3 | 534 |
| 8 | Last three digits form a number divisible by 8 | 5,120 (120) |
| 9 | Sum of digits is divisible by 9 | 6,534 (18) |
| 10 | Last digit is 0 | 7,890 |
| 11 | Difference of the sums of alternate digits is 0 or a multiple of 11 | 8,151 |
| 25 | Last two digits are 00, 25, 50 or 75 | 4,375 |
Working the harder ones
For 11, take 8,151. Digits in odd places from the right: 1 + 1 = 2. Digits in even places: 5 + 8 = 13. The difference is 11, a multiple of 11, so the number is divisible by 11.
For a composite divisor, check the co-prime factors. For 12, check divisibility by 3 and 4, not by 2 and 6, because 2 and 6 share a factor and would pass numbers that fail.
Using them in a primary classroom
- Simplifying 84/126: both digit sums are divisible by 3, so cancel by 3 to get 28/42, then by 7 and 2.
- Checking whether 36 sweets can be shared equally among 9 children before starting to divide.
- Sorting number cards into 'divisible by 3' and 'not divisible by 3' as a game builds the rule better than stating it.
✗ For divisibility by 6, check divisibility by 2 or by 3. | ✓ It must be divisible by both 2 and 3.
✗ For divisibility by 12, check 2 and 6. | ✓ Check 3 and 4, which are co-prime; 2 and 6 are not.
संकेत: 3 से विभाज्यता = अंकों का योग 3 से विभाज्य; 9 से विभाज्यता = अंकों का योग 9 से विभाज्य।
Divisibility supplies about one question, either directly ('which of these is divisible by 11?') or hidden inside a factor, fraction or HCF problem. The two rules most often confused are those for 3 and 9, both using the digit sum but with different divisors, and the rule for 11, which needs alternate sums. The trap on 6 offers 'divisible by 2 or 3'; the answer requires both. Practise the 8 rule, since candidates often check only the last two digits.
Frequently asked questions
What is the divisibility rule for 11?
Add the digits in the odd positions and the digits in the even positions separately, counting from the right. If the difference between the two sums is zero or a multiple of eleven, the number is divisible by 11.
How do you check divisibility by 12?
Check that the number is divisible by both 3 and 4, since these are co-prime and their product is 12. Checking by 2 and 6 is not valid because those two share a common factor.
What is the difference between the rules for 3 and 9?
Both use the sum of the digits. For divisibility by 3 the digit sum must be divisible by 3, and for divisibility by 9 the digit sum must be divisible by 9. Every number divisible by 9 is also divisible by 3.
60-second recap
- 2: last digit even. 5: last digit 0 or 5. 10: last digit 0.
- 3 and 9: digit sum divisible by 3 or by 9 respectively.
- 4: last two digits. 8: last three digits. 25: last two digits 00, 25, 50, 75.
- 6 needs both 2 and 3; 12 needs 3 and 4, which are co-prime.
- 11: difference of alternate digit sums is 0 or a multiple of 11.
Ab isi chapter ke questions CTET Paper 1 mock test me practice karo — number system ke sawaal wahin milenge.