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Solution - Least common multiple (LCM) by prime factorization

1,008
1,008

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 14

Tree view of the prime factors of 14: 2 and 7

The prime factors of 14 are 2 and 7.

3. 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.

4. Find the prime factors of 16

Tree view of the prime factors of 16: 2, 2, 2 and 2

The prime factors of 16 are 2, 2, 2 and 2.

5. Build a prime factors table

Determine the maximum number of times each prime factor (2, 3, 7) occurs in the factorization of the given numbers:

Prime factorNumber12 14 18 16 Max. occurrence
221144
310202
701001

The prime factor 7 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 = 2222337

LCM = 24327

LCM = 1,008

The least common multiple of 12, 14, 18 and 16 is 1,008.

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.