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Solution - Least common multiple (LCM) by prime factorization

2,016
2,016

Step-by-step explanation

1. Find the prime factors of 56

Tree view of the prime factors of 56: 2, 2, 2 and 7

The prime factors of 56 are 2, 2, 2 and 7.

2. 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.

3. Find the prime factors of 84

Tree view of the prime factors of 84: 2, 2, 3 and 7

The prime factors of 84 are 2, 2, 3 and 7.

4. Find the prime factors of 96

Tree view of the prime factors of 96: 2, 2, 2, 2, 2 and 3

The prime factors of 96 are 2, 2, 2, 2, 2 and 3.

5. Build a prime factors table

Determine the maximum number of times each prime factor (2, 3, 7) occurs in the factorization of the given numbers:

Prime factorNumber56 72 84 96 Max. occurrence
233255
302112
710101

The prime factor 7 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 = 22222337

LCM = 25327

LCM = 2,016

The least common multiple of 56, 72, 84 and 96 is 2,016.

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.