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Solution - Least common multiple (LCM) by prime factorization

13,536
13,536

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 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 564

Tree view of the prime factors of 564: 2, 2, 3 and 47

The prime factors of 564 are 2, 2, 3 and 47.

5. Build a prime factors table

Determine the maximum number of times each prime factor (2, 3, 47) occurs in the factorization of the given numbers:

Prime factorNumber24 32 36 564 Max. occurrence
235225
310212
4700011

The prime factor 47 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 = 222223347

LCM = 253247

LCM = 13,536

The least common multiple of 24, 32, 36 and 564 is 13,536.

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.