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Solution - Least common multiple (LCM) by prime factorization

192
192

Step-by-step explanation

1. Find the prime factors of 16

Tree view of the prime factors of 16: 2, 2, 2 and 2

The prime factors of 16 are 2, 2, 2 and 2.

2. Find the prime factors of 32

Tree view of the prime factors of 32: 2, 2, 2, 2 and 2

The prime factors of 32 are 2, 2, 2, 2 and 2.

3. 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.

4. Find the prime factors of 64

Tree view of the prime factors of 64: 2, 2, 2, 2, 2 and 2

The prime factors of 64 are 2, 2, 2, 2, 2 and 2.

5. Build a prime factors table

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

Prime factorNumber16 32 48 64 Max. occurrence
245466
300101

The prime factor 3 occurs 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 = 2222223

LCM = 263

LCM = 192

The least common multiple of 16, 32, 48 and 64 is 192.

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.