what is the least common multiple of 7 and 3

2025

2 min read 31-03-2025

what is the least common multiple of 7 and 3

Finding the least common multiple (LCM) is a fundamental concept in mathematics, particularly useful when working with fractions and simplifying expressions. This article explains how to find the LCM of 7 and 3, and provides a broader understanding of the concept.

Understanding Least Common Multiple

The least common multiple (LCM) of two or more integers is the smallest positive integer that is divisible by all the integers. In simpler terms, it's the smallest number that both (or all) of the numbers can divide into evenly.

For example, let's consider the numbers 2 and 3. Multiples of 2 are 2, 4, 6, 8, 10, and so on. Multiples of 3 are 3, 6, 9, 12, 15, and so on. The smallest number that appears in both lists is 6. Therefore, the LCM of 2 and 3 is 6.

Calculating the LCM of 7 and 3

The numbers 7 and 3 are relatively prime, meaning they share no common factors other than 1. This simplifies the process of finding their LCM considerably. When numbers are relatively prime, their LCM is simply their product.

Therefore, the LCM of 7 and 3 is 7 * 3 = 21.

Methods for Finding the LCM

While the method above works perfectly for relatively prime numbers like 7 and 3, here are two general methods for finding the LCM of any two numbers:

1. Listing Multiples

This method is straightforward but can be time-consuming for larger numbers. List the multiples of each number until you find the smallest multiple common to both lists:

Multiples of 7: 7, 14, 21, 28, 35...
Multiples of 3: 3, 6, 9, 12, 15, 18, 21, 24...

The smallest number appearing in both lists is 21.

2. Prime Factorization

This method is more efficient for larger numbers. It involves finding the prime factorization of each number, then taking the highest power of each prime factor present in either factorization:

Prime factorization of 7: 7 (7 is a prime number)
Prime factorization of 3: 3 (3 is a prime number)

Since there are no common prime factors, the LCM is simply the product of the two numbers: 7 * 3 = 21.

Applications of LCM

The LCM has several applications in mathematics and other fields, including:

Adding and subtracting fractions: Finding a common denominator when adding or subtracting fractions involves finding the LCM of the denominators.
Solving problems involving cycles: For example, if two events occur cyclically, the LCM helps determine when they will occur simultaneously.
Scheduling: The LCM can be used to find the shortest time interval when certain tasks or events repeat.

Conclusion

The least common multiple of 7 and 3 is 21. Understanding how to calculate the LCM is crucial for various mathematical operations and problem-solving scenarios. Whether you use the listing multiples method or the prime factorization method, the result remains the same: the smallest positive number divisible by both 7 and 3 is 21. Remember, for relatively prime numbers, the LCM is simply their product, making the calculation even faster.