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Solution - Least common multiple (LCM) by prime factorization

7,128
7,128

Step-by-step explanation

1. Find the prime factors of 54

Tree view of the prime factors of 54: 2, 3, 3 and 3

The prime factors of 54 are 2, 3, 3 and 3.

2. Find the prime factors of 72

Tree view of the prime factors of 72: 2, 2, 2, 3 and 3

The prime factors of 72 are 2, 2, 2, 3 and 3.

3. Find the prime factors of 81

Tree view of the prime factors of 81: 3, 3, 3 and 3

The prime factors of 81 are 3, 3, 3 and 3.

4. Find the prime factors of 99

Tree view of the prime factors of 99: 3, 3 and 11

The prime factors of 99 are 3, 3 and 11.

5. Build a prime factors table

Determine the maximum number of times each prime factor (2, 3, 11) occurs in the factorization of the given numbers:

Prime factorNumber54 72 81 99 Max. occurrence
213003
332424
1100011

The prime factor 11 occurs one time, while 2 and 3 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 = 222333311

LCM = 233411

LCM = 7,128

The least common multiple of 54, 72, 81 and 99 is 7,128.

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.