What is the Least Common Multiple (LCM) of 17 and 34?
The Least Common Multiple (LCM) is the smallest number that is a multiple of two or more numbers. To find the LCM of 17 and 34, we can use a few different methods.
Method 1: Prime Factorization
-
Find the prime factors of each number:
- 17 is a prime number, so its only prime factor is 17.
- 34 can be factored as 2 x 17.
-
Identify common and unique prime factors:
- Both 17 and 34 share the prime factor 17.
- 34 also has the prime factor 2.
-
Multiply the highest powers of all prime factors:
- The highest power of 2 is 2¹ (from 34).
- The highest power of 17 is 17¹ (from both 17 and 34).
-
Calculate the LCM:
- LCM(17, 34) = 2¹ x 17¹ = 34
Method 2: Listing Multiples
-
List multiples of the larger number (34):
- 34, 68, 102, ...
-
Check if any of these multiples are also multiples of the smaller number (17):
- 34 is also a multiple of 17.
-
The smallest multiple that is shared is the LCM:
- LCM(17, 34) = 34
Understanding LCM
In simple terms, the LCM represents the smallest possible value where both 17 and 34 can divide into it evenly, with no remainder. This is useful in various mathematical contexts, like finding the least common denominator in fractions.
Here's an example:
Let's say you want to add the fractions 1/17 and 1/34. To do that, you need to find a common denominator. The LCM of 17 and 34 is 34, so you would rewrite the fractions as 2/34 and 1/34, respectively.
Conclusion
The Least Common Multiple of 17 and 34 is 34. This can be found using prime factorization, listing multiples, or any other method that helps you find the smallest number divisible by both 17 and 34. Understanding LCM is essential for various mathematical operations, especially when working with fractions.