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Solution - Least common multiple (LCM) by prime factorization

15,750
15,750

Step-by-step explanation

1. Find the prime factors of 75

Tree view of the prime factors of 75: 3, 5 and 5

The prime factors of 75 are 3, 5 and 5.

2. Find the prime factors of 250

Tree view of the prime factors of 250: 2, 5, 5 and 5

The prime factors of 250 are 2, 5, 5 and 5.

3. Find the prime factors of 225

Tree view of the prime factors of 225: 3, 3, 5 and 5

The prime factors of 225 are 3, 3, 5 and 5.

4. Find the prime factors of 525

Tree view of the prime factors of 525: 3, 5, 5 and 7

The prime factors of 525 are 3, 5, 5 and 7.

5. Build a prime factors table

Determine the maximum number of times each prime factor (2, 3, 5, 7) occurs in the factorization of the given numbers:

Prime factorNumber75 250 225 525 Max. occurrence
201001
310212
523223
700011

The prime factors 2 and 7 occur one time, while 3 and 5 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 = 2335557

LCM = 232537

LCM = 15,750

The least common multiple of 75, 250, 225 and 525 is 15,750.

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.