Create reverse_factorial_recursive.py

maths/reverse_factorial_recursive.py

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+def reverse_factorial(n: int, divisor: int = 1) -> int:
+    """
+    Return x such that x! equals the given n, or -1 if no such integer exists.
+
+    This function recursively divides the number `n` by successive integers 
+    starting from 1 until it reaches 1. If the division completes exactly at 1,
+    the last divisor - 1 is the factorial root. Otherwise, return -1.
+
+    Parameters
+    ----------
+    n : int
+        The number whose factorial root is to be found.
+    divisor : int, optional
+        The current divisor used in the recursion (default is 1).
+
+    Returns
+    -------
+    int
+        The factorial root if it exists, otherwise -1.
+
+    Examples
+    --------
+    >>> reverse_factorial(120)
+    5
+    >>> reverse_factorial(24)
+    4
+    >>> reverse_factorial(150)
+    -1
+    >>> reverse_factorial(1)
+    1
+    >>> reverse_factorial(720)
+    6
+    """
+    if n < 1:
+        raise ValueError("Input must be a positive integer.")
+    if n == 1:
+        return divisor
+    if n % divisor != 0:
+        return -1
+    return reverse_factorial(n // divisor, divisor + 1)