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Solution - Least common multiple (LCM) by prime factorization

2,970
2,970

Step-by-step explanation

1. Find the prime factors of 135

Tree view of the prime factors of 135: 3, 3, 3 and 5

The prime factors of 135 are 3, 3, 3 and 5.

2. Find the prime factors of 54

Tree view of the prime factors of 54: 2, 3, 3 and 3

The prime factors of 54 are 2, 3, 3 and 3.

3. Find the prime factors of 110

Tree view of the prime factors of 110: 2, 5 and 11

The prime factors of 110 are 2, 5 and 11.

4. Find the prime factors of 198

Tree view of the prime factors of 198: 2, 3, 3 and 11

The prime factors of 198 are 2, 3, 3 and 11.

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 factorNumber135 54 110 198 Max. occurrence
201111
333023
510101
1100111

The prime factors 2, 5 and 11 occur one time, while 3 occurs 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 = 2333511

LCM = 233511

LCM = 2,970

The least common multiple of 135, 54, 110 and 198 is 2,970.

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.