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Solution - Least common multiple (LCM) by prime factorization

2,880
2,880
See steps

Other Ways to Solve

Least common multiple (LCM) by prime factorization

Step-by-step explanation

1. 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.

2. 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.

3. Find the prime factors of 64

Tree view of the prime factors of 64: 2, 2, 2, 2, 2 and 2

The prime factors of 64 are 2, 2, 2, 2, 2 and 2.

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, 3, 5) occurs in the factorization of the given numbers:

Prime factorNumber45 36 64 80 Max. occurrence
202646
322002
510011

The prime factor 5 occurs 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 = 2⋅2⋅2⋅2⋅2⋅2⋅3⋅3⋅5

LCM = 26⋅32⋅5

LCM = 2,880

The least common multiple of 45, 36, 64 and 80 is 2,880.

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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