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Solution - Least common multiple (LCM) by prime factorization

2,880
2,880
See steps

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Least common multiple (LCM) by prime factorization

Step-by-step explanation

1. Find the prime factors of 24

Tree view of the prime factors of 24: 2, 2, 2 and 3

The prime factors of 24 are 2, 2, 2 and 3.

2. 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.

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 72

Tree view of the prime factors of 72: 2, 2, 2, 3 and 3

The prime factors of 72 are 2, 2, 2, 3 and 3.

5. 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.

6. 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 factorNumber24 40 64 72 120 Max. occurrence
2336336
3100212
5010011

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

7. 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 24, 40, 64, 72 and 120 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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