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Solution - Least common multiple (LCM) by prime factorization

11,340
11,340

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 126

Tree view of the prime factors of 126: 2, 3, 3 and 7

The prime factors of 126 are 2, 3, 3 and 7.

3. 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.

4. Find the prime factors of 252

Tree view of the prime factors of 252: 2, 2, 3, 3 and 7

The prime factors of 252 are 2, 2, 3, 3 and 7.

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 factorNumber81 126 135 252 Max. occurrence
201022
342324
500101
701011

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 = 22333357

LCM = 223457

LCM = 11,340

The least common multiple of 81, 126, 135 and 252 is 11,340.

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.