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Solution - Least common multiple (LCM) by prime factorization

1,050
1,050

Step-by-step explanation

1. Find the prime factors of 3

3 is a prime factor.

2. Find the prime factors of 14

Tree view of the prime factors of 14: 2 and 7

The prime factors of 14 are 2 and 7.

3. Find the prime factors of 70

Tree view of the prime factors of 70: 2, 5 and 7

The prime factors of 70 are 2, 5 and 7.

4. Find the prime factors of 150

Tree view of the prime factors of 150: 2, 3, 5 and 5

The prime factors of 150 are 2, 3, 5 and 5.

5. Build a prime factors table

Determine the maximum number of times each prime factor (2, 3, 5, 7) occurs in the factorization of the given numbers:

Prime factorNumber3 14 70 150 Max. occurrence
201111
310011
500122
701101

The prime factors 2, 3 and 7 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 = 23557

LCM = 23527

LCM = 1,050

The least common multiple of 3, 14, 70 and 150 is 1,050.

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.