DIVISION RULES
Divisibility rules are shortcuts or criteria that help determine if one number is divisible by another without actually performing the division. These rules are useful in arithmetic and number theory. Here are some common divisibility rules for certain divisors:
1. Divisibility by 2:
- Rule: A number is divisible by 2 if its last digit is even (ends in 0, 2, 4, 6, or 8).
2. Divisibility by 3:
- Rule: A number is divisible by 3 if the sum of its digits is divisible by 3.
3. Divisibility by 4:
- Rule: A number is divisible by 4 if the number formed by its last two digits is divisible by 4.
4. Divisibility by 5:
- Rule: A number is divisible by 5 if its last digit is 0 or 5.
5. Divisibility by 6:
- Rule: A number is divisible by 6 if it is divisible by both 2 and 3.
6. Divisibility by 8:
- Rule: A number is divisible by 8 if the number formed by its last three digits is divisible by 8.
7. Divisibility by 9:
- Rule: A number is divisible by 9 if the sum of its digits is divisible by 9.
8. Divisibility by 10:
- Rule: A number is divisible by 10 if it ends in 0.
9. Divisibility by 11:
- Rule: A number is divisible by 11 if the difference between the sum of its odd-positioned digits and the sum of its even-positioned digits is divisible by 11.
10. Divisibility by 12:
- Rule: A number is divisible by 12 if it is divisible by both 3 and 4
Prime numbers are natural numbers greater than 1 that have no positive divisors other than 1 and the number itself. They cannot be formed by multiplying two smaller natural numbers.
Divisibility Rule for Prime Numbers:
A prime number is only divisible by 1 and . It cannot be evenly divided by any other number. This is a fundamental property of prime numbers.
Examples:
Example 1: Prime Number 17
- Divisible by 1? Yes, every number is divisible by 1.
- Divisible by 17? Yes, as 17 is the prime number itself.
- Divisible by any other number? No, it is not divisible by any other number.
Example 2: Prime Number 23
- Divisible by 1? Yes.
- Divisible by 23? Yes, as 23 is the prime number itself.
- Divisible by any other number? No.
Example 3: Prime Number 31
- Divisible by 1? Yes.
- Divisible by 31? Yes, as 31 is the prime number itself.
- Divisible by any other number? No.
Explanation:
The divisibility rule for prime numbers is based on the definition of prime numbers. When we consider any prime number, it is only divisible by 1 and itself. This property is what makes a number prime in the first place.
In general, if you have a prime number , it will be evenly divisible by 1 and , and no other positive integer will divide it evenly. This is a unique characteristic of prime numbers and is a fundamental concept in number theory
| Number | Divisibility Rules |
| 1 | Any number is divisible by 1 |
| 2 | A number is divisible by 2 if its last digit is even |
| 3 | A number is divisible by 3 if the sum of its digits is divisible by 3. |
| 4 | A number is divisible by 4 if the number formed by its last two digits is divisible by 4. |
| 5 | A number is divisible by 5 if its last digit is 0 or 5 |
| 6 | A number is divisible by 6 if it is divisible by both 2 and 3 |
| 7 | For small numbers, a number is divisible by 7 if removing twice the last digit from the remaining truncated number results in a multiple of 7 |
| 8 | A number is divisible by 8 if the number formed by its last three digits is divisible by 8 |
| 9 | A number is divisible by 9 if the sum of its digits is divisible by 9 |
| 10 | A number is divisible by 10 if it ends in 0 |
| 11 | A number is divisible by 11 if the difference between the sum of its odd-positioned digits and the sum of its even-positioned digits is divisible by 11 |
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Solved Examples of Division Rules
Solution: A number is divisible by 2 if its last digit is even. In this case, the last digit of 486 is 6 (even), so 486 is divisible by 2.
Solution: A number is divisible by 3 if the sum of its digits is divisible by 3. For 972, the sum is 9+7+2=18, and since 18 is divisible by 3, 972 is divisible by 3.
Solution: A number is divisible by 4 if the number formed by its last two digits is divisible by 4. In this case, the last two digits are 24, which is divisible by 4, so 8,624 is divisible by 4.
Solution: A number is divisible by 5 if its last digit is 0 or 5. In this case, the last digit of 725 is 5, so 725 is divisible by 5.
Solution: A number is divisible by 9 if the sum of its digits is divisible by 9. For 729, the sum is 7+2+9=18, which is divisible by 9, so 729 is divisible by 9.
Solution: A number is divisible by 11 if the difference between the sum of its odd-positioned digits and the sum of its even-positioned digits is divisible by 11. For 1,287, (1+7)−(2+8)=−2, which is not divisible by 11. Therefore, 1,287 is not divisible by 11. |
