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Solution - Least common multiple (LCM) by prime factorization

600
600
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Least common multiple (LCM) by prime factorization

Step-by-step explanation

1. Find the prime factors of 10

Tree view of the prime factors of 10: 2 and 5

The prime factors of 10 are 2 and 5.

2. Find the prime factors of 20

Tree view of the prime factors of 20: 2, 2 and 5

The prime factors of 20 are 2, 2 and 5.

3. Find the prime factors of 30

Tree view of the prime factors of 30: 2, 3 and 5

The prime factors of 30 are 2, 3 and 5.

4. Find the prime factors of 40

Tree view of the prime factors of 40: 2, 2, 2 and 5

The prime factors of 40 are 2, 2, 2 and 5.

5. Find the prime factors of 50

Tree view of the prime factors of 50: 2, 5 and 5

The prime factors of 50 are 2, 5 and 5.

6. Find the prime factors of 60

Tree view of the prime factors of 60: 2, 2, 3 and 5

The prime factors of 60 are 2, 2, 3 and 5.

7. Build a prime factors table

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

Prime factorNumber10 20 30 40 50 60 Max. occurrence
21213123
30010011
51111212

The prime factor 3 occurs one time, while 2 and 5 occur more than once.

8. Calculate the LCM

The least common multiple is the product of all factors in the greatest number of their occurrence.

LCM = 2⋅2⋅2⋅3⋅5⋅5

LCM = 23⋅3⋅52

LCM = 600

The least common multiple of 10, 20, 30, 40, 50 and 60 is 600.

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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