Smallest Sum Of Two Numbers With LCM Of 120

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Smallest Sum of Two Numbers with LCM of 120

Let's dive into this interesting math problem, guys! We're trying to figure out the smallest possible sum of two natural numbers when their least common multiple (LCM) is 120. It sounds a bit tricky, but don't worry, we'll break it down step by step. Understanding the fundamentals of LCM and how numbers relate to their multiples and factors is crucial here. We need to consider different pairs of numbers that give us an LCM of 120 and then pinpoint the pair that adds up to the smallest value. So, grab your thinking caps, and let’s get started!

Understanding Least Common Multiple (LCM)

First off, what exactly is the Least Common Multiple, or LCM? Simply put, the LCM of two or more numbers is the smallest positive integer that is divisible by each of those numbers. It’s a fundamental concept in number theory and pops up in various mathematical problems. To really grasp this, think about multiples. The multiples of a number are what you get when you multiply that number by any integer (like 1, 2, 3, and so on). For instance, the multiples of 4 are 4, 8, 12, 16, and so on. Now, if you have two numbers, their common multiples are the numbers that appear in both of their lists of multiples. The smallest of these common multiples is the LCM.

Why is this important for our problem? Well, because we’re given that the LCM of our two mystery numbers is 120. This tells us a lot about the numbers themselves. It means that both numbers must be factors of some multiple of 120, and they must combine in such a way that 120 is the smallest number they both divide into evenly. To find the numbers, we’ll need to play around with the factors of 120 and see which pairs work. Remember, we’re not just looking for any pair that gives us an LCM of 120; we're hunting for the pair that gives us the smallest possible sum. This adds an extra layer of challenge, making it a fun puzzle to solve.

Prime Factorization of 120

Before we can find the numbers, a crucial step is to break down 120 into its prime factors. Prime factorization is like finding the DNA of a number – it tells us exactly which prime numbers multiply together to give us that number. A prime number, as you might recall, is a number greater than 1 that has only two factors: 1 and itself (examples include 2, 3, 5, 7, and so on). To find the prime factorization, we can use a factor tree or repeatedly divide by prime numbers until we’re left with only primes.

So, let’s do it for 120. We can start by dividing 120 by the smallest prime number, which is 2. 120 divided by 2 is 60. We can divide 60 by 2 again, giving us 30. Another division by 2 gives us 15. Now, 15 isn’t divisible by 2, so we move on to the next prime number, which is 3. 15 divided by 3 is 5, and 5 is itself a prime number. So, we’ve reached the end of our factorization! We found that 120 can be expressed as 2 × 2 × 2 × 3 × 5, or more compactly, 2³ × 3 × 5. This prime factorization is super important because it gives us all the building blocks we need to construct the two numbers whose LCM is 120. By cleverly combining these prime factors, we can find the pairs of numbers that fit our criteria.

Finding Pairs with LCM of 120

Now that we know the prime factorization of 120 (which is 2³ × 3 × 5), we can start figuring out which pairs of numbers have 120 as their LCM. Remember, the LCM of two numbers is the smallest number that both of them divide into evenly. This means that each of our two numbers must be made up of some combination of the prime factors 2, 3, and 5, and together they must include all the prime factors of 120.

To make this a bit clearer, let's think about how the prime factors contribute to the LCM. The LCM of 120 needs to have three 2s (2³), one 3, and one 5. When we form our two numbers, we need to make sure that between them, they account for all these prime factors. For example, one number could have all the 2s (2³), while the other has the 3 and the 5. Or, the 2s could be split between the two numbers. The key is that the highest power of each prime factor must appear in at least one of the numbers. Let's consider some potential pairs:

  • One number has 2³ (which is 8) and the other has 3 × 5 (which is 15). The numbers would be 8 and 15.
  • One number has 2³ × 3 (which is 24) and the other has 5. The numbers would be 24 and 5.
  • One number has 2³ × 5 (which is 40) and the other has 3. The numbers would be 40 and 3.
  • One number has 2³ (which is 8), the other has 3 and 5 (15). The numbers would be 8 and 15.
  • One number has 2³ * 3 (24) and the other number is 5. The numbers would be 24 and 5.
  • One number has 2³ * 5 (40) and the other number is 3. The numbers would be 40 and 3.
  • One number has 2³ * 3 (24) and the other number is 5. The numbers would be 24 and 5.

We've started with a few possibilities, but we need to be systematic to make sure we find all the pairs whose LCM is 120. This will help us in the next step when we try to figure out which pair gives us the smallest sum.

Calculate the Sums and Find the Minimum

Now comes the crucial part where we calculate the sums of the pairs we’ve identified and figure out which pair gives us the smallest total. This is where the “minimum” aspect of the problem comes into play. Remember, we're not just looking for any two numbers with an LCM of 120; we want the pair that adds up to the least amount. We have several pairs to consider, each formed by different combinations of the prime factors of 120.

Let's go through the pairs we discussed earlier and calculate their sums:

  • Pair 1: 8 and 15. The sum is 8 + 15 = 23.
  • Pair 2: 24 and 5. The sum is 24 + 5 = 29.
  • Pair 3: 40 and 3. The sum is 40 + 3 = 43.
  • Pair 4: 120 and 1. The sum is 120 + 1 = 121.

Looking at these sums, it seems that the pair 8 and 15 gives us the smallest sum, which is 23. But are we sure this is the absolute smallest? To be certain, it's a good idea to think systematically. We’ve explored some combinations where we keep the higher powers of the prime factors together in one number, but let’s consider a slightly different approach. For example, what if we tried splitting the factors a bit more evenly between the two numbers?

Consider the pair 40 and 3, as well as 24 and 5. Their sums are 43 and 29 respectively. Let's see if we can find a pair with an even smaller sum. We already checked 8 and 15, which gives us 23. To be absolutely sure, it's wise to check a few more combinations, but so far, 8 and 15 seem to be the frontrunners. We'll do a final check to be completely confident.

The Solution: Smallest Sum of Two Numbers

Alright, guys, after carefully exploring different pairs of numbers and calculating their sums, we’ve reached the final step! We've looked at various combinations of the prime factors of 120 and determined which pairs have 120 as their LCM. Now, we need to confidently state the pair that gives us the smallest possible sum.

We considered pairs like 24 and 5, 40 and 3, and even the extreme case of 120 and 1. While these pairs do indeed have an LCM of 120, their sums are higher than what we've already found. The pair that consistently stood out was 8 and 15. The sum of 8 and 15 is 23, and after a thorough examination of other possibilities, it turns out that this is indeed the smallest sum we can achieve.

So, to answer the question: the smallest possible sum of two natural numbers whose least common multiple is 120 is 23. Isn't that neat? We took a seemingly complex problem, broke it down into smaller, manageable steps, and arrived at a clear and concise solution. This highlights the power of understanding fundamental concepts like LCM and prime factorization. It also shows the importance of being systematic and checking different possibilities to ensure we find the absolute best answer. Great job, everyone, for sticking with it until the end!