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Solution - Least common multiple (LCM) by prime factorization

2,080
2,080
See steps

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Least common multiple (LCM) by prime factorization

Step-by-step explanation

1. 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.

2. Find the prime factors of 32

Tree view of the prime factors of 32: 2, 2, 2, 2 and 2

The prime factors of 32 are 2, 2, 2, 2 and 2.

3. Find the prime factors of 65

Tree view of the prime factors of 65: 5 and 13

The prime factors of 65 are 5 and 13.

4. Find the prime factors of 80

Tree view of the prime factors of 80: 2, 2, 2, 2 and 5

The prime factors of 80 are 2, 2, 2, 2 and 5.

5. Build a prime factors table

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

Prime factorNumber26 32 65 80 Max. occurrence
215045
500111
1310101

The prime factors 5 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 = 2⋅2⋅2⋅2⋅2⋅5⋅13

LCM = 25⋅5⋅13

LCM = 2,080

The least common multiple of 26, 32, 65 and 80 is 2,080.

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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