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Solution - Least common multiple (LCM) by prime factorization

21,840
21,840

Step-by-step explanation

1. 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.

2. 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.

3. Find the prime factors of 13

13 is a prime factor.

4. Find the prime factors of 16

Tree view of the prime factors of 16: 2, 2, 2 and 2

The prime factors of 16 are 2, 2, 2 and 2.

5. 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.

6. Build a prime factors table

Determine the maximum number of times each prime factor (2, 3, 5, 7, 13) occurs in the factorization of the given numbers:

Prime factorNumber14 10 13 16 21 Max. occurrence
2110404
3000011
5010001
7100011
13001001

The prime factors 3, 5, 7 and 13 occur one time, while 2 occurs more than once.

7. Calculate the LCM

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

LCM = 222235713

LCM = 2435713

LCM = 21,840

The least common multiple of 14, 10, 13, 16 and 21 is 21,840.

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.