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Solution - Least common multiple (LCM) by prime factorization

2,700
2,700

Step-by-step explanation

1. Find the prime factors of 12

Tree view of the prime factors of 12: 2, 2 and 3

The prime factors of 12 are 2, 2 and 3.

2. Find the prime factors of 15

Tree view of the prime factors of 15: 3 and 5

The prime factors of 15 are 3 and 5.

3. 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.

4. 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.

5. 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.

6. Find the prime factors of 150

Tree view of the prime factors of 150: 2, 3, 5 and 5

The prime factors of 150 are 2, 3, 5 and 5.

7. 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 factorNumber12 15 90 108 135 150 Max. occurrence
22012012
31123313
50110122

The prime factors 2, 3 and 5 occur more than once.

8. Calculate the LCM

The least common multiple is the product of all factors in the greatest number of their occurrence.

LCM = 2233355

LCM = 223352

LCM = 2,700

The least common multiple of 12, 15, 90, 108, 135 and 150 is 2,700.

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.