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Solution - Least common multiple (LCM) by prime factorization

4,914
4,914

Step-by-step explanation

1. Find the prime factors of 26

Tree view of the prime factors of 26: 2 and 13

The prime factors of 26 are 2 and 13.

2. Find the prime factors of 54

Tree view of the prime factors of 54: 2, 3, 3 and 3

The prime factors of 54 are 2, 3, 3 and 3.

3. Find the prime factors of 78

Tree view of the prime factors of 78: 2, 3 and 13

The prime factors of 78 are 2, 3 and 13.

4. Find the prime factors of 182

Tree view of the prime factors of 182: 2, 7 and 13

The prime factors of 182 are 2, 7 and 13.

5. Build a prime factors table

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

Prime factorNumber26 54 78 182 Max. occurrence
211111
303103
700011
1310111

The prime factors 2, 7 and 13 occur one time, while 3 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 = 2333713

LCM = 233713

LCM = 4,914

The least common multiple of 26, 54, 78 and 182 is 4,914.

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.