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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 28

Tree view of the prime factors of 28: 2, 2 and 7

The prime factors of 28 are 2, 2 and 7.

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 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.

4. Find the prime factors of 48

Tree view of the prime factors of 48: 2, 2, 2, 2 and 3

The prime factors of 48 are 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 factorNumber28 32 36 48 Max. occurrence
225245
300212
710001

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 28, 32, 36 and 48 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.