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Solution - Least common multiple (LCM) by prime factorization

29,450
29,450

Step-by-step explanation

1. Find the prime factors of 19

19 is a prime factor.

2. Find the prime factors of 31

31 is a prime factor.

3. Find the prime factors of 38

Tree view of the prime factors of 38: 2 and 19

The prime factors of 38 are 2 and 19.

4. Find the prime factors of 25

Tree view of the prime factors of 25: 5 and 5

The prime factors of 25 are 5 and 5.

5. Build a prime factors table

Determine the maximum number of times each prime factor (2, 5, 19, 31) occurs in the factorization of the given numbers:

Prime factorNumber19 31 38 25 Max. occurrence
200101
500022
1910101
3101001

The prime factors 2, 19 and 31 occur one time, while 5 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 = 2551931

LCM = 2521931

LCM = 29,450

The least common multiple of 19, 31, 38 and 25 is 29,450.

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.