Determining the Values of K for Given LCM and GCD Relationships

The least common multiple (LCM) of two numbers, 1728 and K, is 5184. The challenge lies in finding the possible values of K that satisfy this condition. This article explores the method to determine such values using the relationship between LCM, GCD, and prime factorization.

Understand the LCM and GCD Relationship

The LCM of two numbers, a and b, can be written in terms of their GCD as follows:

LCM(a, b) (a × b) / GCD(a, b)

In this problem, we have:

LCM(1728, K) 5184

We can rearrange the formula to solve for K:

K (LCM(1728, K) × GCD(1728, K)) / 1728

Given:

LCM(1728, K) 5184 and 1728 26 × 33

Prime Factorization of Key Numbers

Let's begin by finding the prime factorization of 1728 and 5184:

1728 123 26 × 33

5184 26 × 34

Using the LCM definition:

LCM(1728, K) (1728 × K) / GCD(1728, K) 5184

Therefore, we have:

1728 × K 5184 × GCD(1728, K)

Directly Finding the GCD

To find the GCD, we can use the prime factorizations:

GCD(1728, K) 2min(m) × 3min(n)

Where m and n are the respective exponents of the prime factors in the prime factorization of K. To ensure that the LCM is 5184, we must have:

min(6, a) 6 and min(3, b) 4

This gives us:

a ≤ 6 and b ≥ 4

Possible Values for a and b

The variable a can take any value from 0 to 6, providing 7 possible options (0, 1, 2, 3, 4, 5, 6).

The variable b must be 4, giving us only 1 possible option.

Total number of possible values for K:

7 (for a) × 1 (for b) 7

Conclusion

The total number of possible values of K is 7. These values can be derived by ensuring the GCD and LCM conditions are met. As an example, some possible values for K include: 81, 162, 324, 648, 1296, 2592, and 5184.

Using the GCD and LCM relationships, we can accurately determine the values of K that satisfy the given conditions. This method can be applied to similar problems involving prime factorization and the properties of LCM and GCD.