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Solution - Least common multiple (LCM) by prime factorization

1,440
1,440
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Least common multiple (LCM) by prime factorization

Step-by-step explanation

1. 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.

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 45

Tree view of the prime factors of 45: 3, 3 and 5

The prime factors of 45 are 3, 3 and 5.

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. Find the prime factors of 36

Tree view of the prime factors of 36: 2, 2, 3 and 3

The prime factors of 36 are 2, 2, 3 and 3.

6. Find the prime factors of 120

Tree view of the prime factors of 120: 2, 2, 2, 3 and 5

The prime factors of 120 are 2, 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 factorNumber60 32 45 80 36 120 Max. occurrence
22504235
31020212
51011011

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

LCM = 25⋅32⋅5

LCM = 1,440

The least common multiple of 60, 32, 45, 80, 36 and 120 is 1,440.

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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