Counting Pairs with a Specific LCM: A Comprehensive Guide

When dealing with mathematical problems, particularly in finding pairs of numbers with a specific least common multiple (LCM), it is crucial to understand the underlying principles and steps involved. In this article, we will explore how to determine the number of pairs of two numbers whose LCM is 400, and then generalize the approach for any number. This guide is especially useful for those working in areas like programming, mathematics, and optimization problems.

Understanding the Problem

Given a specific LCM, the task is to find how many distinct pairs of integers can yield that LCM. For this demonstration, we will use the number 400. Let's break down the steps and explore the methods to reach the solution.

Prime Factorization and LCM Relationship

To solve this problem, we start by determining the prime factorization of the given number. For 400, we have:

400  2^4 * 5^2

We are given that the LCM of two numbers, a and b, is 400. Thus, we can express a and b in terms of their prime factors:

a  2^{x_1} * 5^{y_1}b  2^{x_2} * 5^{y_2}

The relationship between the LCM of the two numbers and their greatest common divisor (GCD) is:

LCM(a, b)  frac{a * b}{GCD(a, b)}

Given that LCM(a, b) 400, the exponents x1 and x2, as well as y1 and y2, must satisfy the following conditions:

(max{x_1, x_2} 4) for the prime 2.

(max{y_1, y_2} 2) for the prime 5.

Step-by-Step Calculation

Step 1: Determine the pairs for the factors of 2.

The condition max{x1, x2} 4 indicates that one of the exponents must be 4. The possible values for the other exponent are:

0, 1, 2, 3, or 4 (for x1) 0, 1, 2, 3, or 4 (for x2)

This gives us 5 options for x1 and 5 options for x2, totaling 25 pairs. However, since the scenario where both exponents are 4 is counted twice, we subtract one:

25 - 1 24 pairs.

Step 2: Determine the pairs for the factors of 5.

The condition max{y1, y2} 2 indicates that one of the exponents must be 2. The possible values for the other exponent are:

0 or 1 (for y1) 0 or 1 (for y2)

This gives us 3 options for y1 and 3 options for y2, totaling 9 pairs. Again, we subtract one pair:

9 - 1 8 pairs.

Step 3: Calculate the total number of pairs.

Finally, we multiply the total combinations for the factors of 2 and 5:

24 * 8 192

However, from the initial analysis, we can simplify this manually to:

Total pairs 9 (for 2) * 5 (for 5) 45

Generalization for Any Number

Let's generalize the approach for any number. Consider n 40000000. The prime factorization is:

40000000  2^8 * 5^6

Following the same steps, we need to find the number of pairs whose LCM is 40000000. The exponents for the prime factors 2 and 5 must satisfy:

(max{x_1, x_2} 8) for the prime 2.

(max{y_1, y_2} 6) for the prime 5.

Step 1: Determine the pairs for the factors of 2.

Total pairs 9 (for 2) * 9 (for 5) 81

Step 2: Determine the pairs for the factors of 5.

Total pairs 7 (for 2) * 7 (for 5) 49

Step 3: Calculate the total number of pairs.

Final total pairs 81 * 49 3969

Conclusion

In summary, the process involves determining the prime factorization of the LCM, and then calculating the number of valid pairs using the conditions derived from the maxima of the exponents. The approach can be generalized for any number by following the same steps, making it a powerful tool for solving similar problems in mathematics and related fields.

Additional Questions

Question: How many pairs of 2 numbers are there whose LCM is 40000000?

Answer: 3969

If you have any further questions or need assistance with specific examples, please feel free to comment below!