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 mathematics, crucial for simplifying calculations and understanding number properties. While a calculator can quickly tell you if a number is divisible by 6, understanding the underlying rules empowers you with a deeper mathematical understanding and faster mental arithmetic. This article will explore the divisibility rule for 6 and provide a comprehensive guide to identifying numbers divisible by 6, addressing common misconceptions and providing practical applications.

Understanding Divisibility Rules: The Foundation

Before diving into the specifics of divisibility by 6, let's establish a foundation by understanding the general concept of divisibility rules. A divisibility rule is a shortcut to determine if a number is perfectly divisible by another number without performing long division. These rules are based on the properties of numbers and their relationships.

For example:

• Divisibility by 2: A number is divisible by 2 if its last digit is an even number (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.
• Divisibility by 5: A number is divisible by 5 if its last digit is either 0 or 5.
• Divisibility by 10: A number is divisible by 10 if its last digit is 0.

These rules significantly streamline the process of determining divisibility, especially for larger numbers.

The Divisibility Rule for 6: A Combination of Rules

The divisibility rule for 6 is a composite rule, meaning it combines the divisibility rules for two other numbers: 2 and 3. A number is divisible by 6 only if it's divisible by both 2 and 3. This is because 6 is the product of 2 and 3 (6 = 2 x 3). Therefore, to determine if a number is divisible by 6, we must apply both the divisibility rule for 2 and the divisibility rule for 3.

Step-by-Step Process: Checking Divisibility by 6

Let's break down the process into clear steps:

1. Check for Divisibility by 2: Examine the last digit of the number. If the last digit is even (0, 2, 4, 6, or 8), the number is divisible by 2.

2. Check for Divisibility by 3: Add up all the digits of the number. If the sum of the digits is divisible by 3, then the number itself is divisible by 3.

3. Combine the Results: If the number passes both tests (divisible by 2 and divisible by 3), then it is divisible by 6. If it fails either test, it's not divisible by 6.

Example 1: Is 126 divisible by 6?

1. Divisibility by 2: The last digit is 6 (even), so it's divisible by 2.
2. Divisibility by 3: 1 + 2 + 6 = 9. 9 is divisible by 3, so 126 is divisible by 3.
3. Conclusion: Since 126 is divisible by both 2 and 3, it's divisible by 6.

Example 2: Is 245 divisible by 6?

1. Divisibility by 2: The last digit is 5 (odd), so it's not divisible by 2.
2. Divisibility by 3: 2 + 4 + 5 = 11. 11 is not divisible by 3.
3. Conclusion: Since 245 fails both tests, it's not divisible by 6.

Example 3: Is 312 divisible by 6?

1. Divisibility by 2: The last digit is 2 (even), so it's divisible by 2.
2. Divisibility by 3: 3 + 1 + 2 = 6. 6 is divisible by 3, so 312 is divisible by 3.
3. Conclusion: Since 312 is divisible by both 2 and 3, it's divisible by 6.

Common Mistakes and Misconceptions

A frequent error is to mistakenly believe that if a number is divisible by either 2 or 3, it's automatically divisible by 6. This is incorrect. Both conditions must be met for a number to be divisible by 6.

Another common mistake is incorrectly calculating the sum of the digits when checking divisibility by 3. Careless addition can lead to inaccurate conclusions. Always double-check your addition to avoid errors.

Practical Applications and Real-World Examples

Understanding divisibility by 6 extends beyond theoretical mathematics. It finds practical applications in various areas:

• Inventory Management: If you're arranging items in groups of 6, knowing divisibility by 6 helps determine if you have a complete set or if you'll have any leftovers.

• Problem Solving: Many mathematical word problems rely on understanding divisibility to find solutions efficiently. For example, problems involving sharing items equally among six people require applying divisibility by 6.

• Data Organization: When organizing data into tables or spreadsheets with six columns, divisibility by 6 helps determine how many rows are needed to avoid incomplete rows.

• Game Design: Games often use modular arithmetic (arithmetic with remainders), and understanding divisibility can be crucial for creating fair and balanced gameplay mechanics.

Advanced Concepts and Extensions

The concept of divisibility expands beyond simple whole numbers. It extends to:

• Negative Numbers: The divisibility rules apply equally to negative numbers. A negative number is divisible by 6 if its absolute value is divisible by 6.

• Fractions and Decimals: Fractions and decimals aren't directly divisible in the same way as integers. However, concepts of divisibility are important when working with fractions and decimals in relation to their integer components. For instance, determining if the numerator of a fraction is divisible by 6 can simplify the fraction.

Conclusion: Mastering Divisibility for Mathematical Fluency

The ability to quickly and accurately determine if a number is divisible by 6 is a valuable skill. Understanding the divisibility rule for 6—its reliance on both divisibility by 2 and 3—provides a powerful tool for simplifying calculations and solving problems across various mathematical contexts. By mastering this rule, you enhance your mathematical fluency and confidence in tackling more complex problems. Regular practice and applying these rules in diverse situations will solidify your understanding and increase your efficiency. Remember, the key is not just to memorize the rules but to understand the underlying reasons why they work. This deeper comprehension will empower you to confidently solve divisibility problems and expand your overall mathematical proficiency.