Understanding the Least Common Multiple (LCM) Through Prime Factorization
The Least Common Multiple (LCM) is a fundamental concept in mathematics that is particularly useful in various applications, including simplifying fractions and solving equations. This article will explore the method of finding the LCM of the numbers 42, 45, and 50 using prime factorization. We'll break down the numbers into their prime factors and then determine their LCM step-by-step.
Introduction to Prime Factorization
Prime factorization is the process of expressing a number as the product of its prime factors. A prime number is a number greater than 1 that has no positive divisors other than 1 and itself. The prime factorization of a number is unique, except for the order of the factors.
Prime Factorization of Given Numbers
Let's begin by finding the prime factorization of the numbers 42, 45, and 50.
Prime Factorization of 42
42 can be divided by 2 (the smallest prime number):
42 ÷ 2 21
21 can be further divided by 3:
21 ÷ 3 7
7 is a prime number. Therefore, the prime factorization of 42 is:
42 2 × 3 × 7
Prime Factorization of 45
45 can be divided by 3:
45 ÷ 3 15
15 can be further divided by 3:
15 ÷ 3 5
5 is a prime number. Therefore, the prime factorization of 45 is:
45 32 × 5
Prime Factorization of 50
50 can be divided by 2:
50 ÷ 2 25
25 can be further divided by 5:
25 ÷ 5 5
5 is a prime number. Therefore, the prime factorization of 50 is:
50 2 × 52
Finding the LCM Using Prime Factorization
The Least Common Multiple (LCM) of several numbers is the smallest positive integer that is divisible by each of the numbers without leaving a remainder. To find the LCM, we take the highest power of each prime factor that appears in the prime factorization of the given numbers.
Step-by-Step Method
First, list the prime factors with their highest powers from each number:
42 21 × 31 × 71 45 32 × 51 50 21 × 52Next, take the highest power of each prime factor:
2: The highest power is 21 (from 42 and 50). 3: The highest power is 32 (from 45). 5: The highest power is 52 (from 50). 7: The highest power is 71 (from 42).Finally, multiply these factors together to get the LCM:
LCM 21 × 32 × 52 × 71
Let's perform the multiplication step-by-step:
2 × 32 2 × 9 18 18 × 52 18 × 25 450 450 × 7 3150Therefore, the Least Common Multiple (LCM) of 42, 45, and 50 is:
LCM 3150
Conclusion
Understanding the process of prime factorization and LCM calculation is crucial for various mathematical and practical applications. The LCM of 42, 45, and 50 is 3150, which can be determined by finding the highest powers of all prime factors present in the factorization of the given numbers.
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