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Solution - Least common multiple (LCM) by prime factorization

3,552
3,552

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 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.

4. Find the prime factors of 74

Tree view of the prime factors of 74: 2 and 37

The prime factors of 74 are 2 and 37.

5. Build a prime factors table

Determine the maximum number of times each prime factor (2, 3, 37) occurs in the factorization of the given numbers:

Prime factorNumber24 32 48 74 Max. occurrence
235415
310101
3700011

The prime factors 3 and 37 occur one time, while 2 occurs 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 = 25337

LCM = 3,552

The least common multiple of 24, 32, 48 and 74 is 3,552.

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.