Which Number Is Divisible By Both 3 And 4

Which Numbers are Divisible by Both 3 and 4? A Comprehensive Guide

Finding numbers divisible by both 3 and 4 might seem simple, but understanding the underlying principles reveals a fascinating interplay of number theory. This guide will delve into the methods for identifying these numbers, explaining the concepts clearly and providing practical examples. This will help you understand divisibility rules and how they apply to more complex scenarios involving multiple divisors.

Understanding Divisibility Rules

Before we tackle numbers divisible by both 3 and 4, let's refresh our understanding of individual divisibility rules:

• Divisibility by 3: A number is divisible by 3 if the sum of its digits is divisible by 3. For example, 12 (1+2=3) is divisible by 3, while 14 (1+4=5) is not.

• Divisibility by 4: A number is divisible by 4 if its last two digits are divisible by 4. For example, 104 (04 is divisible by 4) is divisible by 4, but 106 is not.

Finding Numbers Divisible by Both 3 and 4

To find numbers divisible by both 3 and 4, we need to satisfy both divisibility rules simultaneously. This means the number must meet the criteria for both divisibility by 3 and divisibility by 4. There are a few ways to approach this:

Method 1: Testing Individual Numbers

We can systematically test numbers. Let's start with small numbers:

• 12: The sum of digits (1+2=3) is divisible by 3, and the last two digits (12) are divisible by 4. Therefore, 12 is divisible by both 3 and 4.

• 24: The sum of digits (2+4=6) is divisible by 3, and the last two digits (24) are divisible by 4. Therefore, 24 is divisible by both 3 and 4.

• 36: The sum of digits (3+6=9) is divisible by 3, and the last two digits (36) are divisible by 4. Therefore, 36 is divisible by both 3 and 4.

This method becomes less efficient for larger numbers.

Method 2: Finding the Least Common Multiple (LCM)

A more efficient method involves finding the least common multiple (LCM) of 3 and 4. The LCM is the smallest number that is a multiple of both 3 and 4.

• The multiples of 3 are: 3, 6, 9, 12, 15, 18, 21, 24, ...
• The multiples of 4 are: 4, 8, 12, 16, 20, 24, ...

The smallest number that appears in both lists is 12. Therefore, the LCM of 3 and 4 is 12.

Any multiple of the LCM (12) will be divisible by both 3 and 4. So, the numbers divisible by both 3 and 4 are multiples of 12: 12, 24, 36, 48, 60, and so on.

Conclusion

Determining which numbers are divisible by both 3 and 4 can be achieved through direct testing or, more efficiently, by finding the least common multiple (LCM) of 3 and 4, which is 12. All multiples of 12 will satisfy both divisibility rules. Understanding these methods provides a strong foundation for tackling more complex divisibility problems and enhances your understanding of number theory concepts.