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Solution - Least common multiple (LCM) by prime factorization

1,152
1,152

Step-by-step explanation

1. 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.

2. 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.

3. Find the prime factors of 128

Tree view of the prime factors of 128: 2, 2, 2, 2, 2, 2 and 2

The prime factors of 128 are 2, 2, 2, 2, 2, 2 and 2.

4. Find the prime factors of 144

Tree view of the prime factors of 144: 2, 2, 2, 2, 3 and 3

The prime factors of 144 are 2, 2, 2, 2, 3 and 3.

5. Build a prime factors table

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

Prime factorNumber32 96 128 144 Max. occurrence
255747
301022

The prime factors 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 = 222222233

LCM = 2732

LCM = 1,152

The least common multiple of 32, 96, 128 and 144 is 1,152.

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.