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Solution - Least common multiple (LCM) by prime factorization

25,935
25,935

Step-by-step explanation

1. Find the prime factors of 7

7 is a prime factor.

2. Find the prime factors of 13

13 is a prime factor.

3. 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.

4. Find the prime factors of 57

Tree view of the prime factors of 57: 3 and 19

The prime factors of 57 are 3 and 19.

5. Build a prime factors table

Determine the maximum number of times each prime factor (3, 5, 7, 13, 19) occurs in the factorization of the given numbers:

Prime factorNumber7 13 15 57 Max. occurrence
300111
500101
710001
1301001
1900011

The prime factors 3, 5, 7, 13 and 19 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 = 3571319

LCM = 25,935

The least common multiple of 7, 13, 15 and 57 is 25,935.

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.