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Solution - Least common multiple (LCM) by prime factorization

2,052
2,052

Step-by-step explanation

1. 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.

2. Find the prime factors of 76

Tree view of the prime factors of 76: 2, 2 and 19

The prime factors of 76 are 2, 2 and 19.

3. 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.

4. Find the prime factors of 114

Tree view of the prime factors of 114: 2, 3 and 19

The prime factors of 114 are 2, 3 and 19.

5. Build a prime factors table

Determine the maximum number of times each prime factor (2, 3, 19) occurs in the factorization of the given numbers:

Prime factorNumber54 76 108 114 Max. occurrence
212212
330313
1901011

The prime factor 19 occurs 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 = 2233319

LCM = 223319

LCM = 2,052

The least common multiple of 54, 76, 108 and 114 is 2,052.

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.