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Solution - Least common multiple (LCM) by prime factorization

2,520
2,520

Step-by-step explanation

1. Find the prime factors of 18

Tree view of the prime factors of 18: 2, 3 and 3

The prime factors of 18 are 2, 3 and 3.

2. Find the prime factors of 30

Tree view of the prime factors of 30: 2, 3 and 5

The prime factors of 30 are 2, 3 and 5.

3. 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.

4. Find the prime factors of 21

Tree view of the prime factors of 21: 3 and 7

The prime factors of 21 are 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 factorNumber18 30 40 21 Max. occurrence
211303
321012
501101
700011

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 = 2223357

LCM = 233257

LCM = 2,520

The least common multiple of 18, 30, 40 and 21 is 2,520.

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.