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Solution - Least common multiple (LCM) by prime factorization

29,792
29,792

Step-by-step explanation

1. Find the prime factors of 49

Tree view of the prime factors of 49: 7 and 7

The prime factors of 49 are 7 and 7.

2. Find the prime factors of 19

19 is a prime factor.

3. 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.

4. 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.

5. Build a prime factors table

Determine the maximum number of times each prime factor (2, 7, 19) occurs in the factorization of the given numbers:

Prime factorNumber49 19 16 32 Max. occurrence
200455
720002
1901001

The prime factor 19 occurs one time, while 2 and 7 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 = 222227719

LCM = 257219

LCM = 29,792

The least common multiple of 49, 19, 16 and 32 is 29,792.

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.