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Solution - Least common multiple (LCM) by prime factorization

9,639
9,639

Step-by-step explanation

1. 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.

2. 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.

3. Find the prime factors of 51

Tree view of the prime factors of 51: 3 and 17

The prime factors of 51 are 3 and 17.

4. Find the prime factors of 81

Tree view of the prime factors of 81: 3, 3, 3 and 3

The prime factors of 81 are 3, 3, 3 and 3.

5. Build a prime factors table

Determine the maximum number of times each prime factor (3, 7, 17) occurs in the factorization of the given numbers:

Prime factorNumber21 27 51 81 Max. occurrence
313144
710001
1700101

The prime factors 7 and 17 occur one time, while 3 occurs 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 = 3333717

LCM = 34717

LCM = 9,639

The least common multiple of 21, 27, 51 and 81 is 9,639.

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.