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Solution - Least common multiple (LCM) by prime factorization

120
120

Step-by-step explanation

1. Find the prime factors of 15

Tree view of the prime factors of 15: 3 and 5

The prime factors of 15 are 3 and 5.

2. Find the prime factors of 30

Tree view of the prime factors of 30: 2, 3 and 5

The prime factors of 30 are 2, 3 and 5.

3. Find the prime factors of 60

Tree view of the prime factors of 60: 2, 2, 3 and 5

The prime factors of 60 are 2, 2, 3 and 5.

4. Find the prime factors of 120

Tree view of the prime factors of 120: 2, 2, 2, 3 and 5

The prime factors of 120 are 2, 2, 2, 3 and 5.

5. Build a prime factors table

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

Prime factorNumber15 30 60 120 Max. occurrence
201233
311111
511111

The prime factors 3 and 5 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 = 22235

LCM = 2335

LCM = 120

The least common multiple of 15, 30, 60 and 120 is 120.

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.