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Solution - Least common multiple (LCM) by prime factorization

8,400
8,400

Step-by-step explanation

1. Find the prime factors of 35

Tree view of the prime factors of 35: 5 and 7

The prime factors of 35 are 5 and 7.

2. Find the prime factors of 48

Tree view of the prime factors of 48: 2, 2, 2, 2 and 3

The prime factors of 48 are 2, 2, 2, 2 and 3.

3. 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.

4. Find the prime factors of 100

Tree view of the prime factors of 100: 2, 2, 5 and 5

The prime factors of 100 are 2, 2, 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 factorNumber35 48 42 100 Max. occurrence
204124
301101
510022
710101

The prime factors 3 and 7 occur one time, while 2 and 5 occur 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 = 22223557

LCM = 243527

LCM = 8,400

The least common multiple of 35, 48, 42 and 100 is 8,400.

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.