Understanding divisibility rules helps simplify many mathematical problems, especially in number theory and arithmetic. Let’s explore the rules for 5, 6, and 11 in detail.
Divisibility Rule for 5
A number is divisible by 5 if its last digit is either 0 or 5. This rule is straightforward and easy to apply.
Examples:
- 35 is divisible by 5 because its last digit is 5.
- 120 is divisible by 5 because its last digit is 0.
- 123 is not divisible by 5 because its last digit is 3.
Divisibility Rule for 6
A number is divisible by 6 if it meets two criteria: it must be divisible by both 2 and 3.
Step-by-Step Check:
- Divisibility by 2: A number is divisible by 2 if its last digit is even (0, 2, 4, 6, or 8).
- Divisibility by 3: A number is divisible by 3 if the sum of its digits is divisible by 3.
Examples:
- 24 is divisible by 6 because it is even (divisible by 2) and the sum of its digits (2 + 4 = 6) is divisible by 3.
- 36 is divisible by 6 because it is even and the sum of its digits (3 + 6 = 9) is divisible by 3.
- 25 is not divisible by 6 because, although it is divisible by 5, it is not even.
Divisibility Rule for 11
A number is divisible by 11 if the difference between the sum of the digits in odd positions and the sum of the digits in even positions is either 0 or a multiple of 11.
Step-by-Step Check:
- Identify positions: Assign positions to each digit, starting from the rightmost digit as position 1, the next as position 2, and so on.
- Sum of digits in odd positions: Add the digits in odd positions.
- Sum of digits in even positions: Add the digits in even positions.
- Calculate the difference: Subtract the sum of the digits in even positions from the sum of the digits in odd positions.
- Check the result: If the result is 0 or a multiple of 11, the number is divisible by 11.
Examples:
- 121 is divisible by 11 because (1 + 1) – 2 = 0.
- 2728 is divisible by 11 because (2 + 2) – (7 + 8) = 4 – 15 = -11, which is a multiple of 11.
- 1234 is not divisible by 11 because (1 + 3) – (2 + 4) = 4 – 6 = -2, which is not a multiple of 11.
Conclusion
Using these simple rules, you can quickly determine whether a number is divisible by 5, 6, or 11. These rules are handy tools for simplifying calculations and solving problems more efficiently.
3. Wikipedia – Divisibility Rule