Which Of The Following Is Divisible By 6

Which of the Following is Divisible by 6? A Deep Dive into Divisibility Rules

Determining divisibility is a fundamental concept in arithmetic, crucial for simplifying calculations and understanding number properties. While a calculator can quickly verify divisibility, understanding the underlying rules empowers you to solve problems efficiently and build a stronger mathematical foundation. This article focuses specifically on divisibility by 6, exploring the rules, providing examples, and offering advanced techniques for determining divisibility in various contexts.

Understanding Divisibility by 6

A number is divisible by 6 if it's perfectly divisible by both 2 and 3. This means that when you divide the number by 6, the remainder is 0. This seemingly simple rule combines two distinct divisibility tests, making it a powerful tool in number theory. Let's break down the individual tests and then see how they combine to determine divisibility by 6.

Divisibility by 2: The Even Number Check

The rule for divisibility by 2 is straightforward: a number is divisible by 2 if it's an even number, meaning its last digit is 0, 2, 4, 6, or 8. This is because even numbers are multiples of 2. For example:

• 12 is divisible by 2 (ends in 2)
• 100 is divisible by 2 (ends in 0)
• 746 is divisible by 2 (ends in 6)
• 1,235 is not divisible by 2 (ends in 5)

Divisibility by 3: The Sum of Digits Test

The rule for divisibility by 3 is slightly more involved: a number is divisible by 3 if the sum of its digits is divisible by 3. This relies on the concept of modular arithmetic. Let's look at some examples:

• 12: 1 + 2 = 3, and 3 is divisible by 3, so 12 is divisible by 3.
• 102: 1 + 0 + 2 = 3, and 3 is divisible by 3, so 102 is divisible by 3.
• 747: 7 + 4 + 7 = 18, and 18 is divisible by 3 (1 + 8 = 9, which is divisible by 3), so 747 is divisible by 3.
• 1,234: 1 + 2 + 3 + 4 = 10, and 10 is not divisible by 3, so 1,234 is not divisible by 3.

Combining the Tests: Divisibility by 6

Since a number must be divisible by both 2 and 3 to be divisible by 6, we combine the two tests. To determine if a number is divisible by 6:

1. Check for Divisibility by 2: Is the last digit even (0, 2, 4, 6, or 8)? If not, the number is not divisible by 6.
2. Check for Divisibility by 3: If the last digit is even, add up the digits of the number. Is the sum divisible by 3? If yes, the number is divisible by 6. If no, the number is not divisible by 6.

Let's apply this to some examples:

• 102: The last digit (2) is even. The sum of the digits (1 + 0 + 2 = 3) is divisible by 3. Therefore, 102 is divisible by 6.
• 746: The last digit (6) is even. The sum of the digits (7 + 4 + 6 = 17) is not divisible by 3. Therefore, 746 is not divisible by 6.
• 1260: The last digit (0) is even. The sum of the digits (1 + 2 + 6 + 0 = 9) is divisible by 3. Therefore, 1260 is divisible by 6.
• 2345: The last digit (5) is odd, so the number is not divisible by 2 and therefore not divisible by 6.

Advanced Techniques and Applications

Understanding divisibility by 6 extends beyond simple numerical checks. Here are some advanced applications and techniques:

Working with Larger Numbers

For extremely large numbers, repeatedly applying the sum-of-digits rule for divisibility by 3 can simplify the process. For instance:

Consider the number 9,876,543,210. Instead of summing all the digits directly, repeatedly sum the digits until you reach a single-digit number:

9 + 8 + 7 + 6 + 5 + 4 + 3 + 2 + 1 + 0 = 45

4 + 5 = 9

Since 9 is divisible by 3, and the last digit (0) is even, the original number is divisible by 6.

Divisibility and Factorization

Divisibility by 6 is intimately linked to prime factorization. Since 6 = 2 x 3, any number divisible by 6 must contain at least one factor of 2 and at least one factor of 3 in its prime factorization. This connection is valuable in number theory and algebra.

Problem Solving and Real-World Applications

The divisibility rules, including that for 6, are not merely abstract concepts. They are crucial in various contexts:

• Inventory Management: Determining if a certain quantity of items can be evenly distributed into groups of 6.
• Scheduling: Dividing tasks or time slots into 6-hour segments.
• Coding and Algorithms: Optimizing code for efficiency, especially in scenarios involving modular arithmetic.

Common Mistakes to Avoid

While the divisibility rules for 6 are relatively straightforward, some common pitfalls can lead to errors:

• Forgetting to check both divisibility by 2 and 3: Remember, a number must satisfy both conditions to be divisible by 6. Checking only one condition isn't sufficient.
• Incorrectly summing digits: Carefully add the digits of the number to check for divisibility by 3. Even a small mistake in addition will lead to an incorrect result.
• Misunderstanding the order of operations: Apply the divisibility by 2 check first; it's a quick filter. Only proceed to the divisibility by 3 check if the number passes the divisibility by 2 check.

Conclusion

Divisibility by 6, while seemingly a simple arithmetic concept, provides a valuable insight into number properties and offers practical applications in various fields. Mastering this rule, by understanding the underlying divisibility rules for 2 and 3, allows for efficient problem-solving and strengthens your overall mathematical skills. By combining the individual tests and avoiding common pitfalls, you can accurately and confidently determine the divisibility of any number by 6. Remember to practice regularly to build fluency and confidence in your mathematical abilities. This will be valuable in solving more complex mathematical problems later.