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Solution - Least common multiple (LCM) by prime factorization

9,828
9,828

Step-by-step explanation

1. Find the prime factors of 39

Tree view of the prime factors of 39: 3 and 13

The prime factors of 39 are 3 and 13.

2. 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.

3. Find the prime factors of 108

Tree view of the prime factors of 108: 2, 2, 3, 3 and 3

The prime factors of 108 are 2, 2, 3, 3 and 3.

4. Find the prime factors of 26

Tree view of the prime factors of 26: 2 and 13

The prime factors of 26 are 2 and 13.

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 factorNumber39 91 108 26 Max. occurrence
200212
310303
701001
1311011

The prime factors 7 and 13 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 = 22333713

LCM = 2233713

LCM = 9,828

The least common multiple of 39, 91, 108 and 26 is 9,828.

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.