LCM of Two Numbers

by Lily, Aug 02 2023

What is the Least Common Multiple (LCM)?

Multiples: When we multiply a number, like 5, by another number, say 4, we get a result, which is the multiple of 5. For example, 5 x 4 = 20, so 20 is a multiple of 5. It's like the multiplication table we all know and love!
Common Multiples: Now, imagine we list the multiples of two numbers, say 4 and 5. The common multiples are the numbers that appear in both lists. For example, the multiples of 4 are 4, 8, 12, 16, 20, and so on, while the multiples of 5 are 5, 10, 15, 20, and so on. The common multiples are 20, 40, 60, and so forth.
Least Common Multiple (LCM): The LCM is the smallest positive number that is a common multiple of two or more numbers. It's like finding the tiniest superhero of multiples! In other words, the LCM is a number that is a multiple of both numbers.

How to Find LCM:

Method 1 - List and Find: One way to find the LCM is to list the multiples of both numbers and look for the smallest common multiple.
Method 2 - Prime Factorization: Another exciting method is the prime factorization method. First, write each number as a product of its prime factors. Then, multiply the highest power of each prime factor together to get the LCM.
Method 3 - Ladder Method: The ladder method is also super cool! Simultaneously divide both numbers by prime numbers until the division is even. Multiply the divisors to get the LCM.

Let's Solve Some LCM Mysteries:

Find the LCM of 4 and 6:

Multiples of 4: 4, 8, 12, 16, 20, ...
Multiples of 6: 6, 12, 18, 24, ...
The LCM of 4 and 6 is 12, the smallest number common in both lists!

Find the LCM of 6 and 18:

Prime factorization of 6: 2 × 3
Prime factorization of 18: 2 × 3 × 3
LCM: 2 × 3 × 3 = 18

Find the LCM of 30 and 45:

Prime factorization of 30: 2 × 3 × 5
Prime factorization of 45: 3 × 3 × 5
LCM: 2 × 3 × 3 × 5 = 90

The Differences Between GCF and LCM:

Remember, LCM is all about finding the smallest common multiple, while the Greatest Common Factor (GCF) is the highest divisor of the numbers.

Fun Facts:

Euclid, the mathematician, introduced the concept of LCM.
LCM is not only used in math but also in encoding messages and cryptography.
It helps optimize quantities under operations.

Conclusion

The LCM is the smallest of the common multiples of two or more numbers. Multiples are numbers you get by multiplying a number by another, and common multiples are the ones that appear in both lists.

To find the LCM, we can use different methods like listing multiples, finding prime factors, or using the ladder method. For example, to find the LCM of 3 and 5, we list their multiples and find the smallest common multiple, which is 15. We can also find the LCM of bigger numbers by breaking them down into prime factors and then multiplying the highest power of each prime.

LCM is important in various fields, including cryptography and optimization. Have fun exploring the LCM magic!