Enter an equation or problem
Camera input is not recognized!

Solution - Least common multiple (LCM) by prime factorization

3,780
3,780

Step-by-step explanation

1. Find the prime factors of 42

Tree view of the prime factors of 42: 2, 3 and 7

The prime factors of 42 are 2, 3 and 7.

2. Find the prime factors of 60

Tree view of the prime factors of 60: 2, 2, 3 and 5

The prime factors of 60 are 2, 2, 3 and 5.

3. Find the prime factors of 84

Tree view of the prime factors of 84: 2, 2, 3 and 7

The prime factors of 84 are 2, 2, 3 and 7.

4. Find the prime factors of 108

Tree view of the prime factors of 108: 2, 2, 3, 3 and 3

The prime factors of 108 are 2, 2, 3, 3 and 3.

5. Build a prime factors table

Determine the maximum number of times each prime factor (2, 3, 5, 7) occurs in the factorization of the given numbers:

Prime factorNumber42 60 84 108 Max. occurrence
212222
311133
501001
710101

The prime factors 5 and 7 occur one time, while 2 and 3 occur more than once.

6. Calculate the LCM

The least common multiple is the product of all factors in the greatest number of their occurrence.

LCM = 2233357

LCM = 223357

LCM = 3,780

The least common multiple of 42, 60, 84 and 108 is 3,780.

Why learn this

The least common multiple (LCM), sometimes called the lowest common multiple or least common divisor, is helpful for understanding the relationships between numbers. For example, if it takes Earth 365 days to orbit the sun and it takes Venus 225 days to orbit the sun and both are in perfect alignment at the time this scenario is given, how long will it take for Earth and Venus to align again? We can use LCM to determine that the answer would be 16,425 days.

LCM is also a very important part of many mathematical concepts that also have real-world applications. For example, we use LCMs when adding and subtracting fractions, which we use quite frequently.