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Solution - Least common multiple (LCM) by prime factorization

6,615
6,615

Step-by-step explanation

1. Find the prime factors of 27

Tree view of the prime factors of 27: 3, 3 and 3

The prime factors of 27 are 3, 3 and 3.

2. 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.

3. Find the prime factors of 45

Tree view of the prime factors of 45: 3, 3 and 5

The prime factors of 45 are 3, 3 and 5.

4. Find the prime factors of 49

Tree view of the prime factors of 49: 7 and 7

The prime factors of 49 are 7 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 factorNumber27 35 45 49 Max. occurrence
330203
501101
701022

The prime factor 5 occurs one time, while 3 and 7 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 = 333577

LCM = 33572

LCM = 6,615

The least common multiple of 27, 35, 45 and 49 is 6,615.

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.