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Solution - Least common multiple (LCM) by prime factorization

2,100
2,100

Step-by-step explanation

1. Find the prime factors of 25

Tree view of the prime factors of 25: 5 and 5

The prime factors of 25 are 5 and 5.

2. Find the prime factors of 28

Tree view of the prime factors of 28: 2, 2 and 7

The prime factors of 28 are 2, 2 and 7.

3. Find the prime factors of 14

Tree view of the prime factors of 14: 2 and 7

The prime factors of 14 are 2 and 7.

4. Find the prime factors of 42

Tree view of the prime factors of 42: 2, 3 and 7

The prime factors of 42 are 2, 3 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 factorNumber25 28 14 42 Max. occurrence
202112
300011
520002
701111

The prime factors 3 and 7 occur 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 = 223557

LCM = 223527

LCM = 2,100

The least common multiple of 25, 28, 14 and 42 is 2,100.

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.