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Solution - Least common multiple (LCM) by prime factorization

13,500
13,500

Step-by-step explanation

1. Find the prime factors of 540

Tree view of the prime factors of 540: 2, 2, 3, 3, 3 and 5

The prime factors of 540 are 2, 2, 3, 3, 3 and 5.

2. 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.

3. Find the prime factors of 225

Tree view of the prime factors of 225: 3, 3, 5 and 5

The prime factors of 225 are 3, 3, 5 and 5.

4. Find the prime factors of 750

Tree view of the prime factors of 750: 2, 3, 5, 5 and 5

The prime factors of 750 are 2, 3, 5, 5 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 factorNumber540 108 225 750 Max. occurrence
222012
333213
510233

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 = 22333555

LCM = 223353

LCM = 13,500

The least common multiple of 540, 108, 225 and 750 is 13,500.

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.