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Solution - Least common multiple (LCM) by prime factorization

228,150
228,150
See steps

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Least common multiple (LCM) by prime factorization

Step-by-step explanation

1. Find the prime factors of 338

Tree view of the prime factors of 338: 2, 13 and 13

The prime factors of 338 are 2, 13 and 13.

2. Find the prime factors of 702

Tree view of the prime factors of 702: 2, 3, 3, 3 and 13

The prime factors of 702 are 2, 3, 3, 3 and 13.

3. Find the prime factors of 675

Tree view of the prime factors of 675: 3, 3, 3, 5 and 5

The prime factors of 675 are 3, 3, 3, 5 and 5.

4. Find the prime factors of 975

Tree view of the prime factors of 975: 3, 5, 5 and 13

The prime factors of 975 are 3, 5, 5 and 13.

5. Build a prime factors table

Determine the maximum number of times each prime factor (2, 3, 5, 13) occurs in the factorization of the given numbers:

Prime factorNumber338 702 675 975 Max. occurrence
211001
303313
500222
1321012

The prime factor 2 occurs one time, while 3, 5 and 13 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 = 2⋅3⋅3⋅3⋅5⋅5⋅13⋅13

LCM = 2⋅33⋅52⋅132

LCM = 228,150

The least common multiple of 338, 702, 675 and 975 is 228,150.

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.

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