How to Find Common Multiples of 12 and 8 Between 50 and 100

Finding common multiples can be a useful mathematical operation, especially when dealing with specific ranges. In this article, we'll walk through the process of finding the common multiples of 12 and 8 that fall between 50 and 100. This involves determining the least common multiple (LCM) of the two numbers and then identifying which multiples of that LCM fall within the specified range.

Step 1: Calculating the LCM of 12 and 8

Before finding the common multiples, it's essential to determine the least common multiple (LCM) of the two numbers, 12 and 8. The LCM is the smallest positive integer that is divisible by the given numbers.

Prime Factorization

To find the LCM, we can use the prime factorization method. This involves breaking down each number into its prime factors.

The prime factorization of 12 is:

12 22 × 3

The prime factorization of 8 is:

8 23

Now, we take the highest power of each prime number involved in the factorizations:

For 2, the highest power is 23. For 3, the highest power is 31.

Therefore, the LCM of 12 and 8 is:

(text{LCM}(12, 8) 2^3 times 3^1 8 times 3 24)

Step 2: Finding Multiples of 24 Between 50 and 100

Now that we have the LCM, we need to find the multiples of 24 that lie within the range of 50 to 100.

The multiples of 24 are:

24, 48, 72, 96, 120, ...

Simplifying the range for our purposes:

24 x 1 24 (not in range) 24 x 2 48 (not in range) 24 x 3 72 (in range) 24 x 4 96 (in range) 24 x 5 120 (not in range)

The multiples of 24 that fall between 50 and 100 are 72 and 96.

Counting Common Multiples

The common multiples of 12 and 8 between 50 and 100 are:

72 96

So, there are 2 common multiples of 12 and 8 between 50 and 100.

Additional Insight

Why not just pick up a pencil and a piece of paper and count the multiples? It can indeed be a simple task. The multiples of 12 greater than 50 and less than 100 are 60, 72, 84, and 96. All of these numbers are multiples of 12, and as expected, 72 and 96 are also multiples of 8, making them the common multiples.

Conclusion

Understanding the least common multiple (LCM) and how to find common multiples within a specified range is an important skill in mathematics, particularly in applications like scheduling or engineering. By breaking down the problem into logical steps and using prime factorization, we can efficiently solve these types of problems.