Explaining The Least Common Multiple Of 12 And 5

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Understanding the Concept of Least Common Multiple

Before we delve into the least common multiple of 12 and 5, let us first understand the concept of least common multiple. In mathematics, the least common multiple is the smallest positive integer that is divisible by two or more given numbers without leaving any remainder. For instance, the least common multiple of 4 and 6 is 12 since 12 is the smallest number that is a multiple of both 4 and 6. Similarly, the least common multiple of 3 and 5 is 15 since 15 is the smallest number that is a multiple of both 3 and 5.

Calculating the Least Common Multiple of 12 and 5

To calculate the least common multiple of 12 and 5, we need to find the multiples of both numbers and then identify the smallest multiple that occurs in both lists. Let us first find the multiples of 12 and 5. Multiples of 12: 12, 24, 36, 48, 60, 72, 84, 96, 108, 120, 132, 144, 156, 168, 180... Multiples of 5: 5, 10, 15, 20, 25, 30, 35, 40, 45, 50, 55, 60, 65, 70, 75, 80, 85, 90, 95, 100, 105, 110, 115, 120, 125, 130, 135, 140, 145, 150, 155, 160, 165, 170, 175, 180... As we can see, the smallest number that appears in both lists is 60. Therefore, the least common multiple of 12 and 5 is 60.

Using Prime Factorization to Find the Least Common Multiple

Another way to find the least common multiple of two numbers is by using prime factorization. To use this method, we need to first find the prime factors of both 12 and 5. Prime factors of 12: 2, 2, 3 Prime factors of 5: 5 Next, we need to find the highest power of each prime factor that occurs in either number. In this case, the highest power of 2 is 2^2, and the highest power of 3 is 3^1 since these are the only prime factors that appear in 12. For 5, we only have one prime factor, which is 5^1. Finally, we multiply the highest powers of each prime factor to get the least common multiple. In this case, the least common multiple is 2^2 x 3^1 x 5^1, which simplifies to 60.

Why is Finding the Least Common Multiple Important?

Finding the least common multiple is important in many areas of mathematics, including fractions and ratios. For instance, when adding or subtracting fractions with different denominators, we need to find the least common multiple of the denominators to make the fractions equivalent. Similarly, when simplifying ratios, we need to find the least common multiple of the terms in the ratio to express it in its simplest form. Thus, understanding how to find the least common multiple is essential for many mathematical operations.

Conclusion

In conclusion, the least common multiple of 12 and 5 is 60. We can find the least common multiple by either listing the multiples of both numbers and identifying the smallest number that appears in both lists or by using prime factorization to find the highest powers of each prime factor and multiplying them together. Understanding the concept of least common multiple is important in various areas of mathematics, making it an essential skill to learn.