What Number Is Divisible By 3 And 4

What Numbers Are Divisible by 3 and 4? A Deep Dive into Divisibility Rules and Number Theory

Finding numbers divisible by both 3 and 4 might seem like a simple arithmetic problem, but it opens a fascinating door into the world of number theory and divisibility rules. This exploration will delve into the fundamental concepts, providing you with not just the answer but a comprehensive understanding of the underlying principles. We'll explore various approaches, from simple trial-and-error to leveraging powerful mathematical tools. By the end, you'll be able to confidently identify any number divisible by both 3 and 4, and understand why.

Understanding Divisibility Rules

Before diving into numbers divisible by both 3 and 4, let's refresh our understanding of individual divisibility rules. These rules provide efficient ways to determine if a number is divisible by a specific integer without performing long division.

Divisibility Rule for 3

A number is divisible by 3 if the sum of its digits is divisible by 3. For example:

• 123: 1 + 2 + 3 = 6 (divisible by 3), therefore 123 is divisible by 3.
• 456: 4 + 5 + 6 = 15 (divisible by 3), therefore 456 is divisible by 3.
• 789: 7 + 8 + 9 = 24 (divisible by 3), therefore 789 is divisible by 3.
• 100: 1 + 0 + 0 = 1 (not divisible by 3), therefore 100 is not divisible by 3.

Divisibility Rule for 4

A number is divisible by 4 if its last two digits form a number divisible by 4. For example:

• 1204: The last two digits, 04, are divisible by 4, therefore 1204 is divisible by 4.
• 3456: The last two digits, 56, are divisible by 4, therefore 3456 is divisible by 4.
• 7892: The last two digits, 92, are divisible by 4, therefore 7892 is divisible by 4.
• 1010: The last two digits, 10, are not divisible by 4, therefore 1010 is not divisible by 4.

Finding Numbers Divisible by Both 3 and 4

Since a number must satisfy both conditions to be divisible by both 3 and 4, we need to combine these divisibility rules. This means a number must simultaneously:

1. Have a sum of digits divisible by 3.
2. Have its last two digits divisible by 4.

Let's illustrate this with examples:

• 108:

  • Sum of digits: 1 + 0 + 8 = 9 (divisible by 3)
  • Last two digits: 08 (divisible by 4)
  • Therefore, 108 is divisible by both 3 and 4.

• 312:

  • Sum of digits: 3 + 1 + 2 = 6 (divisible by 3)
  • Last two digits: 12 (divisible by 4)
  • Therefore, 312 is divisible by both 3 and 4.

• 432:

  • Sum of digits: 4 + 3 + 2 = 9 (divisible by 3)
  • Last two digits: 32 (divisible by 4)
  • Therefore, 432 is divisible by both 3 and 4.

• 500:

  • Sum of digits: 5 + 0 + 0 = 5 (not divisible by 3)
  • Last two digits: 00 (divisible by 4)
  • Therefore, 500 is NOT divisible by both 3 and 4.

• 612:

  • Sum of digits: 6 + 1 + 2 = 9 (divisible by 3)
  • Last two digits: 12 (divisible by 4)
  • Therefore, 612 is divisible by both 3 and 4.

The Least Common Multiple (LCM) Approach

A more sophisticated approach involves the concept of the Least Common Multiple (LCM). The LCM of two numbers is the smallest positive number that is divisible by both numbers. Since we are interested in numbers divisible by both 3 and 4, we need to find the LCM of 3 and 4.

The prime factorization of 3 is simply 3. The prime factorization of 4 is 2 x 2 = 2².

To find the LCM, we take the highest power of each prime factor present in either number: 2² x 3 = 12.

Therefore, the LCM of 3 and 4 is 12. This means any number divisible by both 3 and 4 is also divisible by 12. This provides a more efficient way to identify such numbers. All multiples of 12 (12, 24, 36, 48, 60, and so on) will be divisible by both 3 and 4.

Applying the LCM in Practice

Let's use the LCM method to check some numbers:

• Is 72 divisible by both 3 and 4? 72 / 12 = 6 (a whole number), so yes.
• Is 84 divisible by both 3 and 4? 84 / 12 = 7 (a whole number), so yes.
• Is 96 divisible by both 3 and 4? 96 / 12 = 8 (a whole number), so yes.
• Is 100 divisible by both 3 and 4? 100 / 12 = 8.333... (not a whole number), so no.

Beyond the Basics: Exploring Number Theory Concepts

This exploration of divisibility by 3 and 4 touches upon several fundamental concepts in number theory:

• Divisibility Rules: These shortcuts streamline the process of determining divisibility without performing long division. Understanding these rules is crucial for efficient calculations and problem-solving.

• Prime Factorization: Breaking down numbers into their prime factors is a cornerstone of number theory. It's essential for understanding LCM and GCD (Greatest Common Divisor).

• Least Common Multiple (LCM): The LCM is a vital concept for finding the smallest number divisible by a set of given numbers. It has broad applications in various mathematical fields.

• Modular Arithmetic: While not explicitly covered here, the concepts of divisibility are closely linked to modular arithmetic, a system of arithmetic for integers, where numbers "wrap around" upon reaching a certain value (the modulus).

Conclusion: Mastering Divisibility

Determining whether a number is divisible by both 3 and 4 isn't just about applying simple rules; it's about understanding the underlying mathematical principles. By combining the individual divisibility rules for 3 and 4, or more efficiently using the LCM of 3 and 4 (which is 12), you can effectively identify any number divisible by both. This knowledge extends beyond simple arithmetic, providing a foundation for more advanced explorations in number theory and related mathematical fields. Remember, the key is not just to find the answer but to understand why the answer is correct – a crucial aspect of mathematical understanding and problem-solving. This deeper comprehension allows for more confident and efficient handling of divisibility problems in various contexts.