Type-Level Greatest Common Divisor (GCD) in TypeScript

Calculating the Greatest Common Divisor (GCD) is a fundamental mathematical operation. This challenge asks you to implement a type-level GCD function in TypeScript, leveraging advanced type manipulations. This allows you to perform GCD calculations at compile time, which can be useful for ensuring type safety and optimizing code that relies on common divisors.

Problem Description

You need to create a TypeScript type GCD<A extends number, B extends number> that calculates the Greatest Common Divisor of two number types A and B at the type level. The GCD is the largest positive integer that divides both A and B without leaving a remainder. Your implementation should utilize recursive type-level programming techniques.

Key Requirements:

• The GCD type must accept two number types as generic arguments.
• The result of GCD<A, B> should be a number type representing the GCD of A and B.
• The implementation should handle the Euclidean algorithm at the type level. The Euclidean algorithm states: GCD(A, B) = GCD(B, A % B) until B is 0. The remainder operation (A % B) needs to be implemented using type-level arithmetic.
• The implementation must correctly handle edge cases, including when one or both inputs are zero. GCD(A, 0) = A and GCD(0, B) = B. GCD(0, 0) = 0.

Expected Behavior:

The type GCD<A, B> should resolve to a number type representing the GCD. For example, GCD<12, 18> should resolve to the type 6.

Edge Cases to Consider:

• One or both inputs are zero.
• One input is a multiple of the other.
• Inputs are relatively prime (GCD is 1).
• Large number inputs (though the practical limit is determined by TypeScript's type inference capabilities).

Examples

Example 1:

Input: GCD<12, 18>
Output: 6
Explanation: The GCD of 12 and 18 is 6.

Example 2:

Input: GCD<18, 12>
Output: 6
Explanation: The GCD of 18 and 12 is 6 (order doesn't matter).

Example 3:

Input: GCD<24, 36>
Output: 12
Explanation: The GCD of 24 and 36 is 12.

Example 4:

Input: GCD<7, 13>
Output: 1
Explanation: 7 and 13 are relatively prime, so their GCD is 1.

Example 5:

Input: GCD<10, 0>
Output: 10
Explanation: GCD(10, 0) = 10

Example 6:

Input: GCD<0, 10>
Output: 10
Explanation: GCD(0, 10) = 10

Example 7:

Input: GCD<0, 0>
Output: 0
Explanation: GCD(0, 0) = 0

Constraints

• A and B must be number types.
• The implementation should be as efficient as possible within the constraints of type-level programming. Excessive recursion could lead to type inference errors.
• The solution should be readable and well-documented.
• The solution should not rely on external libraries.

Notes

• You'll need to implement type-level arithmetic, specifically the modulo operation (%). This will likely involve creating helper types to represent numbers and perform calculations.
• Consider using conditional types and recursion to implement the Euclidean algorithm.
• TypeScript's type inference can be sensitive to the complexity of type-level calculations. Start with simpler cases and gradually build up to more complex ones.
• Think about how to handle the base case of the recursion (when one of the inputs becomes zero).
• This is a challenging problem that requires a good understanding of TypeScript's type system and advanced type manipulation techniques. Good luck!