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Solution - Least common multiple (LCM) by prime factorization

300
300

Step-by-step explanation

1. Find the prime factors of 10

Tree view of the prime factors of 10: 2 and 5

The prime factors of 10 are 2 and 5.

2. Find the prime factors of 12

Tree view of the prime factors of 12: 2, 2 and 3

The prime factors of 12 are 2, 2 and 3.

3. 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.

4. 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.

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 factorNumber10 12 15 75 Max. occurrence
212002
301111
510122

The prime factor 3 occurs one time, while 2 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 = 22355

LCM = 22352

LCM = 300

The least common multiple of 10, 12, 15 and 75 is 300.

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.