Finding The Smallest Sum Of Two Numbers With An LCM Of 45

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Hey guys! Let's dive into a cool math problem. We're asked to find the smallest possible sum of two different natural numbers (positive integers) whose Least Common Multiple (LCM) is 45. This type of question often pops up in math contests and can be a bit tricky if you don't break it down step by step. Don't worry, we'll go through it together, and I'll explain everything in a way that's easy to understand. So, grab your pencils and let's get started!

Understanding the Problem: The Basics

Okay, so the core of this problem revolves around two main concepts: natural numbers and the Least Common Multiple (LCM). Let's make sure we're all on the same page. Natural numbers are simply the positive whole numbers – 1, 2, 3, and so on. The LCM, or Least Common Multiple, of two or more numbers is the smallest positive integer that is divisible by all the numbers. For example, the LCM of 3 and 5 is 15 because 15 is the smallest number that both 3 and 5 divide into evenly. Think of it like this: If you list out the multiples of each number, the LCM is the first number that appears in both lists. This definition is super important, so make sure you understand it!

Now, in our problem, we know that the LCM of two different natural numbers is 45. Our mission is to find those two numbers and then calculate their smallest possible sum. The key here is to realize that since the LCM is 45, both numbers must be factors of 45 or combinations of its factors. Since the numbers are different, we can't use the same factor twice. The smallest sum will be made with the smallest factors that have an LCM of 45. Let's explore how to find these factors and their combinations to find the smallest sum.

Breaking Down the LCM: Prime Factorization

One of the most effective strategies for solving LCM problems is to use prime factorization. Prime factorization means expressing a number as a product of its prime factors. A prime number is a whole number greater than 1 that has only two divisors: 1 and itself (e.g., 2, 3, 5, 7, 11, etc.).

So, let's find the prime factorization of 45. We can break down 45 as follows: 45 = 3 x 15 = 3 x 3 x 5. Therefore, the prime factorization of 45 is 3² x 5. This means that any two numbers with an LCM of 45 must be composed of these prime factors (3 and 5) in some way.

Identifying Potential Number Pairs

Now we've got the prime factorization, we can start to figure out what pairs of numbers could possibly have an LCM of 45. Remember that the LCM must include all the prime factors raised to their highest powers. Also, remember, since the question asks for different numbers, both numbers can't be the same.

Here are a few potential pairs we can consider, making sure that when we calculate their LCM, it results in 45:

  • 1 and 45: The LCM(1, 45) = 45. The sum is 1 + 45 = 46.
  • 5 and 9: The LCM(5, 9) = 45. The sum is 5 + 9 = 14.
  • 3 and 45: The LCM(3, 45) = 45. The sum is 3 + 45 = 48.
  • 15 and 3: The LCM(15, 3) = 15. This pair is not appropriate because the LCM is not 45.

It is important to understand that the LCM is the product of all prime factors raised to their highest power in either number. So in the pair (5, 9), 5 is 5¹ and 9 is 3², and so the LCM is 3² x 5¹ = 45.

Analyzing Possible Sums

Now that we have some potential pairs, let's calculate their sums to see which one is the smallest. From the possibilities we looked at before, we have 1 + 45 = 46 and 5 + 9 = 14. We need to explore a bit more to ensure we have the absolute smallest sum. Let's analyze other combinations of factors of 45.

Remember, the goal is to keep the numbers as small as possible while ensuring their LCM is still 45. We've already explored several combinations, but it's crucial to be methodical to avoid missing a smaller sum. The factors of 45 are 1, 3, 5, 9, 15, and 45. Let's consider these factors and how they can be combined to achieve an LCM of 45, making sure to avoid identical numbers.

For example, if we take 3 and 15, their LCM is 15, which is not what we want. If we take 1 and 45, their sum is 46, but this does not give us the smallest sum. If we consider 5 and 9, the LCM is 45, and the sum is 14. This is a very interesting result and provides us with a small sum already, so we can now narrow our focus on the other factors of 45 and how they combine.

Identifying the Smallest Sum

Based on our analysis, the pair (5, 9) gives us an LCM of 45, and their sum is 14. We must consider other possible pairs before concluding. Other possible pairs would lead to a bigger sum. So, the smallest sum is 14. This is the answer we are looking for.

The Answer

Therefore, the smallest value for the sum of two different natural numbers whose LCM is 45 is 14. So, the correct answer is E) 14. Congratulations, you've successfully solved the problem! You can use this method for similar problems. Good luck with your future math endeavors!