Introduction to Least Common Multiple (LCM)

The least common multiple (LCM) is a fundamental concept in arithmetic that is useful in various mathematical operations, such as adding or subtracting fractions with different denominators. This article will explore how to find the LCM of a set of numbers, including the prime factorization method and the step-by-step approach, with detailed examples to help you understand and apply this concept effectively.

What is the LCM of 5, 7, and 15?

Prime Factorization Method for LCM of 5, 7, and 15

To find the LCM of the numbers 5, 7, and 15, we can follow the prime factorization method:

Prime Factorization 5: $5 5^1$ 7: $7 7^1$ 15: $15 3^1 times 5^1$ Identify the Highest Powers of Each Prime The primes involved are 3, 5, and 7. The highest powers are: $3^1$ from 15 $5^1$ from both 5 and 15 $7^1$ from 7 Multiply the Highest Powers Together $LCM 3^1 times 5^1 times 7^1 3 times 5 times 7 105$

Step-by-Step Approach for LCM of 5, 7, and 15

Choose the Largest Number 15 is the largest number. Check Divisibility 15 is not divisible by all the numbers (5 and 7). Start with the table of the largest number (15): 15 × 2 30 (not divisible by all) 15 × 3 45 (not divisible by all) 15 × 4 60 (not divisible by all) 15 × 5 75 (not divisible by all) 15 × 6 90 (not divisible by all) 15 × 7 105 (divisible by all) Conclusion The LCM of 5, 7, and 15 is 105.

LCM of 5, 6, 7, and 14

In this section, we will find the LCM of 5, 6, 7, and 14 using prime factorization:

Prime Factorization 5: $5 5^1$ 6: $6 2^1 times 3^1$ 7: $7 7^1$ 14: $14 2^1 times 7^1$ Identify Unique Prime Factors Unique prime factors: 2, 3, 5, 7 Exponentiate the Prime Factors Highest exponents: 2^1 3^1 5^1 7^1 Multiply the Prime Factors Together LCM $2^1 times 3^1 times 5^1 times 7^1 2 times 3 times 5 times 7 210$

Conclusion

In summary, the LCM of 5, 6, 7, and 14 is 210. This method, involving prime factorization and the step-by-step approach, is also applicable to any set of numbers. Whether you are working with small numbers like 5, 7, and 15 or larger numbers like 5, 6, 7, and 14, the steps remain the same, making the process easy to follow and understand.