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Solution - Least common multiple (LCM) by prime factorization

140,595
140,595

Step-by-step explanation

1. Find the prime factors of 39

Tree view of the prime factors of 39: 3 and 13

The prime factors of 39 are 3 and 13.

2. Find the prime factors of 65

Tree view of the prime factors of 65: 5 and 13

The prime factors of 65 are 5 and 13.

3. Find the prime factors of 91

Tree view of the prime factors of 91: 7 and 13

The prime factors of 91 are 7 and 13.

4. Find the prime factors of 103

103 is a prime factor.

5. Build a prime factors table

Determine the maximum number of times each prime factor (3, 5, 7, 13, 103) occurs in the factorization of the given numbers:

Prime factorNumber39 65 91 103 Max. occurrence
310001
501001
700101
1311101
10300011

The prime factors 3, 5, 7, 13 and 103 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 = 35713103

LCM = 140,595

The least common multiple of 39, 65, 91 and 103 is 140,595.

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.