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Solution - Least common multiple (LCM) by prime factorization

1,440
1,440

Step-by-step explanation

1. Find the prime factors of 15

Tree view of the prime factors of 15: 3 and 5

The prime factors of 15 are 3 and 5.

2. 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.

3. 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.

4. 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.

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 factorNumber15 24 32 45 Max. occurrence
203505
311022
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 = 22222335

LCM = 25325

LCM = 1,440

The least common multiple of 15, 24, 32 and 45 is 1,440.

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.