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Solution - Least common multiple (LCM) by prime factorization

105,625
105,625
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Least common multiple (LCM) by prime factorization

Step-by-step explanation

1. Find the prime factors of 625

Tree view of the prime factors of 625: 5, 5, 5 and 5

The prime factors of 625 are 5, 5, 5 and 5.

2. Find the prime factors of 105,625

Tree view of the prime factors of 105,625: 5, 5, 5, 5, 13 and 13

The prime factors of 105,625 are 5, 5, 5, 5, 13 and 13.

3. Build a prime factors table

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

Prime factorNumber625 105,625Max. occurrence
5444
13022

The prime factors 5 and 13 occur more than once.

4. Calculate the LCM

The least common multiple is the product of all factors in the greatest number of their occurrence.

LCM = 5⋅5⋅5⋅5⋅13⋅13

LCM = 54⋅132

LCM = 105,625

The least common multiple of 625 and 105,625 is 105,625.

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.

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