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Solution - Least common multiple (LCM) by prime factorization

5,040
5,040

Step-by-step explanation

1. Find the prime factors of 72

Tree view of the prime factors of 72: 2, 2, 2, 3 and 3

The prime factors of 72 are 2, 2, 2, 3 and 3.

2. Find the prime factors of 20

Tree view of the prime factors of 20: 2, 2 and 5

The prime factors of 20 are 2, 2 and 5.

3. Find the prime factors of 28

Tree view of the prime factors of 28: 2, 2 and 7

The prime factors of 28 are 2, 2 and 7.

4. Find the prime factors of 48

Tree view of the prime factors of 48: 2, 2, 2, 2 and 3

The prime factors of 48 are 2, 2, 2, 2 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 factorNumber72 20 28 48 Max. occurrence
232244
320012
501001
700101

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 = 22223357

LCM = 243257

LCM = 5,040

The least common multiple of 72, 20, 28 and 48 is 5,040.

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.