Enter an equation or problem
Camera input is not recognized!

Solution - Least common multiple (LCM) by prime factorization

26,325
26,325

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 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.

3. 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.

4. Find the prime factors of 75

Tree view of the prime factors of 75: 3, 5 and 5

The prime factors of 75 are 3, 5 and 5.

5. Build a prime factors table

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

Prime factorNumber39 81 65 75 Max. occurrence
314014
500122
1310101

The prime factor 13 occurs one time, while 3 and 5 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 = 33335513

LCM = 345213

LCM = 26,325

The least common multiple of 39, 81, 65 and 75 is 26,325.

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.