In computational mathematics and number theory, finding the Kth least number for the expression $(2^x) \times (3^y) \times (5^z)$ is a classic problem involving multiplicative combinations of prime factors. This article explores the essence of this issue with technical insights, practical examples, and provides a structured methodology to solve it efficiently.

Understanding the Expression

The expression $(2^x) \times (3^y) \times (5^z)$ defines a set of numbers that are products of powers of 2, 3, and 5. Each number in this set can be represented by the tuple $(x, y, z)$ where:

• $x$, $y$, $z$ are non-negative integers.
• The number set is generated by varying these integers.

The objective is to find the Kth least number, meaning the Kth smallest number that can be derived from this expression with different values of $x$, $y$, and $z$.

Approach and Algorithm

To solve this problem, a naive approach involves generating all possible numbers from combinations of powers of 2, 3, and 5 and then sorting them to find the Kth smallest. However, this approach is computationally expensive due to the vast number of possibilities.

A more efficient method leverages a min-heap (priority queue) data structure because it allows for efficiently retrieving and removing the smallest element. Here's a step-by-step approach using a min-heap:

1. Initialize the Data Structures:
   • A min-heap is initialized with the smallest element, $2^0 \times 3^0 \times 5^0 = 1$.
   • A set (or hash map) is used to track the elements already in the heap to avoid duplicates.
2. Iterate to Find the Kth Element:
   • Extract the smallest element from the heap.
   • Insert the next possible elements derived by multiplying the current element with 2, 3, and 5, if they haven't been inserted earlier.
   • Repeat the process and extract elements until the Kth element is reached.
3. Return the Kth Element:
   • The element extracted on the Kth iteration is the desired Kth least number.

Example

Consider finding the 7th least number from the expression $(2^x) \times (3^y) \times (5^z)$.

1. Initialize the heap with `[1]`.
2. Extract 1, push $2$, $3$, $5$ into the heap.
3. Extract 2, push $2^2$, $2 \times 3$, $2 \times 5$.
4. Extract 3, push $3^2$, $3 \times 2$, $3 \times 5$.
5. Extract 4, push $4 \times 2$, $4 \times 3$, $4 \times 5$.
6. Extract 5, push $5 \times 2$, $5 \times 3$, $5 \times 5$.
7. Continue extracting and inserting until the 7th extraction.

The sequence of numbers extracted would be `1, 2, 3, 4, 5, 6, 8`, so 8 would be our answer.

Key Considerations and Complexity

• Efficiency:
  • The use of the min-heap ensures that the operation of finding the next smallest element is logarithmic concerning the number of elements in the heap.
• Space Complexity:
  • The space complexity is primarily determined by the heap and set, which can grow significantly based on $K$ given the possible number combinations.
• Edge Cases:
  • Handling large K values requires ensuring the algorithm is optimized to prevent overflow and maximize computational efficiency.

Summary Table

Conclusion

Finding the Kth least number from the expression $(2^x) \times (3^y) \times (5^z)$ involves strategic optimizations using efficient data structures. By implementing a min-heap, one can avoid the exhaustive search paradigm and achieve a solution that is computationally feasible for practical applications. This problem not only underscores the importance of algorithm design but also serves as an archetype for solving problems involving prime factors and numeric sequences.