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Solution - Least common multiple (LCM) by prime factorization

16,200
16,200
See steps

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Least common multiple (LCM) by prime factorization

Step-by-step explanation

1. Find the prime factors of 81

Tree view of the prime factors of 81: 3, 3, 3 and 3

The prime factors of 81 are 3, 3, 3 and 3.

2. Find the prime factors of 90

Tree view of the prime factors of 90: 2, 3, 3 and 5

The prime factors of 90 are 2, 3, 3 and 5.

3. Find the prime factors of 100

Tree view of the prime factors of 100: 2, 2, 5 and 5

The prime factors of 100 are 2, 2, 5 and 5.

4. Find the prime factors of 120

Tree view of the prime factors of 120: 2, 2, 2, 3 and 5

The prime factors of 120 are 2, 2, 2, 3 and 5.

5. Build a prime factors table

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

Prime factorNumber81 90 100 120 Max. occurrence
201233
342014
501212

The prime factors 2, 3 and 5 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⋅3⋅3⋅5⋅5

LCM = 23⋅34⋅52

LCM = 16,200

The least common multiple of 81, 90, 100 and 120 is 16,200.

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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