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Solution - Least common multiple (LCM) by prime factorization

105
105

Step-by-step explanation

1. Find the prime factors of 7

7 is a prime factor.

2. 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.

3. Find the prime factors of 21

Tree view of the prime factors of 21: 3 and 7

The prime factors of 21 are 3 and 7.

4. Find the prime factors of 35

Tree view of the prime factors of 35: 5 and 7

The prime factors of 35 are 5 and 7.

5. Build a prime factors table

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

Prime factorNumber7 15 21 35 Max. occurrence
301101
501011
710111

The prime factors 3, 5 and 7 occur one time.

6. Calculate the LCM

The least common multiple is the product of all factors in the greatest number of their occurrence.

LCM = 357

LCM = 105

The least common multiple of 7, 15, 21 and 35 is 105.

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.