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Solution - Least common multiple (LCM) by prime factorization

3,960
3,960
See steps

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Least common multiple (LCM) by prime factorization

Step-by-step explanation

1. Find the prime factors of 22

Tree view of the prime factors of 22: 2 and 11

The prime factors of 22 are 2 and 11.

2. Find the prime factors of 33

Tree view of the prime factors of 33: 3 and 11

The prime factors of 33 are 3 and 11.

3. Find the prime factors of 45

Tree view of the prime factors of 45: 3, 3 and 5

The prime factors of 45 are 3, 3 and 5.

4. 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.

5. Build a prime factors table

Determine the maximum number of times each prime factor (2, 3, 5, 11) occurs in the factorization of the given numbers:

Prime factorNumber22 33 45 72 Max. occurrence
210033
301222
500101
1111001

The prime factors 5 and 11 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 = 2⋅2⋅2⋅3⋅3⋅5⋅11

LCM = 23⋅32⋅5⋅11

LCM = 3,960

The least common multiple of 22, 33, 45 and 72 is 3,960.

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.

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