What is the least common multiple (LCM) of 4 and 7?

To find the least common multiple (LCM) of 4 and 7, we need to identify the smallest number that both 4 and 7 can divide into without leaving a remainder. Let's break this down step by step.

Understanding Multiples

First, let's clarify what a multiple is. A multiple of a number is the product of that number and any integer. For example:

• Multiples of 4: 4, 8, 12, 16, 20, 24, ...
• Multiples of 7: 7, 14, 21, 28, 35, ...

Finding the LCM

Now, we can find the LCM by listing the multiples of each number until we find the smallest common one. Let's look at the multiples we listed:

• Multiples of 4: 4, 8, 12, 16, 20, 24, 28, 32, 36, 40, ...
• Multiples of 7: 7, 14, 21, 28, 35, 42, ...

From these lists, we can see that the first common multiple is 28. Therefore, the LCM of 4 and 7 is 28.

Another Method: Prime Factorization

We can also find the LCM using prime factorization. Here’s how it works:

• 4 can be factored into primes as 2 × 2 (or 2²).
• 7 is already a prime number.

To find the LCM using prime factors, we take the highest power of each prime number that appears in the factorization:

• From 4: 2²
• From 7: 7¹

Now, we multiply these together:

LCM = 2² × 7¹ = 4 × 7 = 28

Summary

In both methods, we arrive at the same result. The least common multiple of 4 and 7 is 28. This number is significant because it represents the smallest quantity that can be evenly divided by both original numbers, making it useful in various mathematical applications, such as finding common denominators in fractions.