Understanding The Least Common Multiple Of 12 And 16

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What is a Multiple?

Before we dive into the concept of the least common multiple of 12 and 16, let's first define what a multiple is. A multiple is a product of a given number and any other whole number. For example, the first few multiples of 12 are 12, 24, 36, 48, and so on.

What is the Least Common Multiple?

Now that we know what a multiple is, let's talk about the least common multiple (LCM). The LCM is the smallest multiple that two or more numbers have in common. In other words, it is the smallest number that is a multiple of both 12 and 16.

How to Find the LCM?

There are different methods to find the LCM, but we will focus on the prime factorization method. To find the LCM of 12 and 16 using this method, we need to follow these steps:

Factor both numbers into their prime factors. 12 can be written as 2 x 2 x 3, while 16 can be written as 2 x 2 x 2 x 2.
Identify the common factors. Both numbers have two 2's, so we write 2 x 2.
Include the remaining factors. We already included the common factors, so we only need to include the 3 from 12 and the remaining 2 from 16. Our final answer is 2 x 2 x 2 x 3, which is 24.

Why is 24 the LCM of 12 and 16?

To understand why 24 is the LCM of 12 and 16, we need to check if it is a multiple of both numbers. We know that 24 is a multiple of 12 because 24 divided by 12 is 2. We also know that 24 is a multiple of 16 because 24 divided by 16 is 1 with a remainder of 8. This means that 24 is the smallest number that is a multiple of both 12 and 16.

What are the Other Multiples?

While 24 is the LCM of 12 and 16, there are other multiples that these two numbers have in common. For example, the next few multiples of 24 are 48, 72, 96, and so on. These are also multiples of both 12 and 16.

Why is LCM Useful?

Knowing the LCM of two or more numbers is useful in many mathematical problems. For example, if we want to add or subtract fractions with different denominators, we need to find their least common denominator, which is the same as the LCM of their denominators. The LCM is also important in finding the period of repeating decimals.

Conclusion

In summary, the least common multiple (LCM) is the smallest multiple that two or more numbers have in common. To find the LCM of 12 and 16, we can use the prime factorization method and get 24 as our answer. Knowing the LCM is useful in various mathematical problems and helps us simplify computations.