Maximum Sum Of Two Different Natural Numbers With LCM 60

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What is the maximum sum of two different natural numbers whose least common multiple (LCM) is 60?

In this comprehensive article, we will delve into an intriguing mathematical problem: determining the maximum possible sum of two distinct natural numbers whose least common multiple (LCM) is 60. This problem combines elements of number theory, including factorization, LCM, and the properties of natural numbers. We will explore the underlying concepts, derive a step-by-step solution, and discuss the mathematical principles involved. This exploration aims to provide a clear and detailed understanding, making it accessible for both students and math enthusiasts.

The core of the problem lies in finding two different natural numbers. Let's denote these numbers as a and b, where a ≠ b. The key condition is that the least common multiple (LCM) of a and b must be 60. The LCM of two numbers is the smallest positive integer that is divisible by both numbers. Our objective is to find the pair of numbers a and b that satisfy this condition while maximizing their sum (a + b).

To effectively tackle this problem, we need to understand the concept of LCM and its relationship with the prime factorization of numbers. The LCM is intimately connected with the prime factors of the numbers involved. Therefore, we will begin by examining the prime factorization of 60, which will serve as the foundation for our solution.

The first crucial step in solving this problem is to determine the prime factorization of 60. Prime factorization involves expressing a number as a product of its prime factors. Prime numbers are numbers greater than 1 that have only two divisors: 1 and themselves (e.g., 2, 3, 5, 7, 11, etc.).

To find the prime factorization of 60, we can use a factor tree or successive division by prime numbers. Let's proceed with successive division:

  • 60 ÷ 2 = 30
  • 30 ÷ 2 = 15
  • 15 ÷ 3 = 5
  • 5 ÷ 5 = 1

From this process, we find that the prime factors of 60 are 2, 2, 3, and 5. Therefore, the prime factorization of 60 can be written as:

60 = 22 × 3 × 5

This prime factorization is essential because it provides us with the building blocks for constructing the numbers a and b. The LCM of a and b being 60 implies that the prime factors of a and b, when combined, must include 22, 3, and 5. The challenge now is to distribute these prime factors between a and b in such a way that their sum is maximized.

Now that we have the prime factorization of 60, which is 22 × 3 × 5, we need to figure out how to distribute these factors between two different numbers, a and b, such that their LCM is 60 and their sum (a + b) is maximized. The key here is to understand that the LCM includes the highest power of each prime factor present in either number.

To maximize the sum a + b, we should aim to make one of the numbers as large as possible while still ensuring that the LCM remains 60. This can be achieved by assigning as many prime factors as possible to one number and the remaining factors to the other number, provided that the LCM condition is met.

Let's consider the possible ways to distribute the prime factors:

  1. One number could include all the factors (22 × 3 × 5 = 60), and the other number should not share all factors to ensure they are different but still result in an LCM of 60.
  2. We need to identify a smaller number that, when combined with 60, still yields an LCM of 60. This means the smaller number can only be a factor of 60.

To maximize the sum, we need to choose the smallest factor of 60 that is different from 60 itself. The factors of 60 are 1, 2, 3, 4, 5, 6, 10, 12, 15, 20, 30, and 60. The smallest factor other than 60 is 1. However, if one number is 60 and the other is 1, their LCM is 60, but this might not yield the maximum sum compared to other combinations.

Let's explore another approach. To keep the numbers distinct and maximize their sum, we can make one number a significant factor of 60 and the other number such that it contributes the remaining prime factors to achieve the LCM of 60. For instance, consider these possibilities:

  • Let a = 60. We need to find a b such that LCM(a, b) = 60. If we choose b = any factor of 60 (other than 60), the LCM will still be 60. We want to choose the smallest such b to maximize the sum. The smallest factor other than 60 is 1. So, let's consider 60 and 1. The sum is 61.
  • Now, let's try another approach. We can divide the prime factors somewhat evenly to see if we get a larger sum. Consider splitting the factors such that one number has 22 × 3 = 12 and the other has 5. In this case, let a = 12 and b = 5. The LCM(12, 5) = 60, and the sum is 17. This is smaller than 61.
  • Let's consider another possibility. Let a = 22 × 5 = 20 and b = 3. The LCM(20, 3) = 60, and the sum is 23, still smaller than 61.
  • We can also try a = 3 × 5 = 15 and b = 22 = 4. The LCM(15, 4) = 60, and the sum is 19, which is also smaller than 61.

Based on our exploration of various combinations, we've identified that the combination of 60 and 1 yields the largest sum while maintaining an LCM of 60. Therefore, the numbers are a = 60 and b = 1. Their sum is:

Sum = a + b = 60 + 1 = 61

To ensure that this is indeed the maximum sum, we can analyze other potential combinations. We've already tried dividing the prime factors in different ways, such as 12 and 5, 20 and 3, and 15 and 4, and none of these combinations resulted in a sum greater than 61. This reinforces our conclusion that choosing one number as 60 and the other as 1 maximizes the sum.

Therefore, the maximum sum of two different natural numbers whose LCM is 60 is 61.

In conclusion, to find the maximum sum of two different natural numbers whose LCM is 60, we first determined the prime factorization of 60, which is 22 × 3 × 5. We then explored different ways to distribute these prime factors between two numbers, a and b, ensuring that their LCM remained 60. By considering various combinations and their sums, we found that the combination of 60 and 1 yields the largest sum, which is 61. This solution highlights the importance of understanding prime factorization and LCM in solving number theory problems. The key to maximizing the sum in this case was to make one number as large as possible (60) while choosing the smallest possible different number (1) that still satisfies the LCM condition. This problem provides a valuable exercise in mathematical reasoning and problem-solving, illustrating how fundamental concepts can be applied to solve intriguing questions.

The maximum sum of two different natural numbers whose LCM is 60 is 61.