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Solution - Least common multiple (LCM) by prime factorization

2,184
2,184

Step-by-step explanation

1. 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.

2. Find the prime factors of 56

Tree view of the prime factors of 56: 2, 2, 2 and 7

The prime factors of 56 are 2, 2, 2 and 7.

3. Find the prime factors of 91

Tree view of the prime factors of 91: 7 and 13

The prime factors of 91 are 7 and 13.

4. Find the prime factors of 84

Tree view of the prime factors of 84: 2, 2, 3 and 7

The prime factors of 84 are 2, 2, 3 and 7.

5. Build a prime factors table

Determine the maximum number of times each prime factor (2, 3, 7, 13) occurs in the factorization of the given numbers:

Prime factorNumber14 56 91 84 Max. occurrence
213023
300011
711111
1300101

The prime factors 3, 7 and 13 occur one time, while 2 occurs 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 = 2223713

LCM = 233713

LCM = 2,184

The least common multiple of 14, 56, 91 and 84 is 2,184.

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.