A number is a multiple of 6 if it is divisible by both 2 and 3.
Understanding Multiples of 6
Multiples of 6 are numbers that result from multiplying 6 by any integer. For example, 6 × 1 = 6, 6 × 2 = 12, 6 × 3 = 18, and so on. These numbers form a distinct sequence (6, 12, 18, 24, 30, 36, 42, 48, 54, 60, etc.) where the difference between any two consecutive multiples is consistently 6. Knowing if a number is a multiple of 6 is useful in various mathematical operations, from simplifying fractions to understanding number patterns.
The Divisibility Rule for 6
Since the number 6 can be factored into its prime components as 2 × 3, a number is a multiple of 6 if and only if it satisfies the divisibility rules for both 2 and 3 independently.
Divisibility Rule for 2
A number is divisible by 2 if its last digit (the digit in the ones place) is an even number. Even numbers include 0, 2, 4, 6, and 8.
- Example:
- 24 is divisible by 2 because its last digit is 4 (an even number).
- 35 is not divisible by 2 because its last digit is 5 (an odd number).
Divisibility Rule for 3
A number is divisible by 3 if the sum of its digits is divisible by 3. You can repeat this process if the sum is still a large number.
- Example:
- 126 is divisible by 3 because the sum of its digits (1 + 2 + 6 = 9) is divisible by 3.
- 47 is not divisible by 3 because the sum of its digits (4 + 7 = 11) is not divisible by 3.
Putting It All Together: Examples
To determine if a number is a multiple of 6, you must apply both rules. Only if a number passes both the divisibility test for 2 and the divisibility test for 3, can it be confirmed as a multiple of 6.
Let's look at some examples:
| Number | Divisible by 2? (Last digit even) | Divisible by 3? (Sum of digits divisible by 3) | Is it a Multiple of 6? | Explanation |
|---|---|---|---|---|
| 42 | Yes (2 is even) | Yes (4 + 2 = 6, which is divisible by 3) | Yes | Both conditions met. |
| 72 | Yes (2 is even) | Yes (7 + 2 = 9, which is divisible by 3) | Yes | Both conditions met. |
| 95 | No (5 is odd) | Yes (9 + 5 = 14, not div by 3) | No | Not divisible by 2. |
| 102 | Yes (2 is even) | Yes (1 + 0 + 2 = 3, which is divisible by 3) | Yes | Both conditions met. |
| 123 | No (3 is odd) | Yes (1 + 2 + 3 = 6, which is divisible by 3) | No | Not divisible by 2. |
| 138 | Yes (8 is even) | Yes (1 + 3 + 8 = 12, which is divisible by 3) | Yes | Both conditions met. |
| 150 | Yes (0 is even) | Yes (1 + 5 + 0 = 6, which is divisible by 3) | Yes | Both conditions met. |
For further reading on divisibility rules, you can explore resources like Khan Academy's content on divisibility or Math is Fun's explanation of divisibility tests.
Practical Insights
Understanding the divisibility rule for 6 allows you to quickly determine if a number can be evenly divided by 6 without performing long division. This skill is particularly useful in:
- Mental Math: Quickly checking numbers for divisibility.
- Simplifying Fractions: Identifying common factors to reduce fractions.
- Problem Solving: Efficiently solving number theory problems.
By applying these straightforward rules, you can easily identify multiples of 6 in any given set of numbers.
