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Solution - Least common multiple (LCM) by prime factorization

4,368
4,368

Step-by-step explanation

1. Find the prime factors of 42

Tree view of the prime factors of 42: 2, 3 and 7

The prime factors of 42 are 2, 3 and 7.

2. 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.

3. Find the prime factors of 104

Tree view of the prime factors of 104: 2, 2, 2 and 13

The prime factors of 104 are 2, 2, 2 and 13.

4. Find the prime factors of 112

Tree view of the prime factors of 112: 2, 2, 2, 2 and 7

The prime factors of 112 are 2, 2, 2, 2 and 7.

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 factorNumber42 78 104 112 Max. occurrence
211344
311001
710011
1301101

The prime factors 3, 7 and 13 occur one time, while 2 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 = 22223713

LCM = 243713

LCM = 4,368

The least common multiple of 42, 78, 104 and 112 is 4,368.

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.