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Solution - Least common multiple (LCM) by prime factorization

5,400
5,400

Step-by-step explanation

1. Find the prime factors of 24

Tree view of the prime factors of 24: 2, 2, 2 and 3

The prime factors of 24 are 2, 2, 2 and 3.

2. Find the prime factors of 40

Tree view of the prime factors of 40: 2, 2, 2 and 5

The prime factors of 40 are 2, 2, 2 and 5.

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 450

Tree view of the prime factors of 450: 2, 3, 3, 5 and 5

The prime factors of 450 are 2, 3, 3, 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 factorNumber24 40 135 450 Max. occurrence
233013
310323
501122

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 = 22233355

LCM = 233352

LCM = 5,400

The least common multiple of 24, 40, 135 and 450 is 5,400.

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.