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Solution - Least common multiple (LCM) by prime factorization

540
540

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 27

Tree view of the prime factors of 27: 3, 3 and 3

The prime factors of 27 are 3, 3 and 3.

3. Find the prime factors of 60

Tree view of the prime factors of 60: 2, 2, 3 and 5

The prime factors of 60 are 2, 2, 3 and 5.

4. 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.

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 factorNumber18 27 60 90 Max. occurrence
210212
323123
500111

The prime factor 5 occurs 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 = 223335

LCM = 22335

LCM = 540

The least common multiple of 18, 27, 60 and 90 is 540.

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.