From a24ebd5da38f8f066d13f9824ce95243d8eacb44 Mon Sep 17 00:00:00 2001
From: nickzerjeski <nick@zerjeski.de>
Date: Mon, 13 Apr 2026 10:44:01 +0200
Subject: [PATCH 1/2] feat(maths): add repunit theorem helpers

---
 maths/repunit_theorem.py | 63 ++++++++++++++++++++++++++++++++++++++++
 1 file changed, 63 insertions(+)
 create mode 100644 maths/repunit_theorem.py

diff --git a/maths/repunit_theorem.py b/maths/repunit_theorem.py
new file mode 100644
index 000000000000..2779147b8545
--- /dev/null
+++ b/maths/repunit_theorem.py
@@ -0,0 +1,63 @@
+"""
+Utilities related to repunits and a classical repunit divisibility theorem.
+
+A repunit of length ``k`` is the number made of ``k`` ones:
+``R_k = 11...1``.
+
+For every positive integer ``n`` with ``gcd(n, 10) = 1``,
+there exists a repunit ``R_k`` divisible by ``n``.
+"""
+
+from math import gcd
+
+
+def has_repunit_multiple(divisor: int) -> bool:
+    """
+    Check whether a divisor admits a repunit multiple.
+
+    >>> has_repunit_multiple(7)
+    True
+    >>> has_repunit_multiple(13)
+    True
+    >>> has_repunit_multiple(2)
+    False
+    >>> has_repunit_multiple(25)
+    False
+    """
+    if divisor <= 0:
+        raise ValueError("divisor must be a positive integer")
+    return gcd(divisor, 10) == 1
+
+
+def least_repunit_length(divisor: int) -> int:
+    """
+    Return the smallest length ``k`` such that repunit ``R_k`` is divisible by divisor.
+
+    Uses modular arithmetic to avoid constructing huge integers.
+
+    >>> least_repunit_length(3)
+    3
+    >>> least_repunit_length(7)
+    6
+    >>> least_repunit_length(41)
+    5
+    """
+    if divisor <= 0:
+        raise ValueError("divisor must be a positive integer")
+    if not has_repunit_multiple(divisor):
+        raise ValueError("divisor must be coprime to 10")
+
+    remainder = 0
+    for length in range(1, divisor + 1):
+        remainder = (remainder * 10 + 1) % divisor
+        if remainder == 0:
+            return length
+
+    # Unreachable when gcd(divisor, 10) == 1 (pigeonhole principle theorem).
+    raise ArithmeticError("no repunit length found for divisor")
+
+
+if __name__ == "__main__":
+    import doctest
+
+    doctest.testmod()

From 237a45f6e6c6f0381b2ba7da0e86f7638963b81c Mon Sep 17 00:00:00 2001
From: Nick Zerjeski <57059725+nickzerjeski@users.noreply.github.com>
Date: Tue, 14 Apr 2026 10:16:16 +0200
Subject: [PATCH 2/2] Update maths/repunit_theorem.py

Co-authored-by: Copilot <175728472+Copilot@users.noreply.github.com>
---
 maths/repunit_theorem.py | 4 ++++
 1 file changed, 4 insertions(+)

diff --git a/maths/repunit_theorem.py b/maths/repunit_theorem.py
index 2779147b8545..8f6757063cc8 100644
--- a/maths/repunit_theorem.py
+++ b/maths/repunit_theorem.py
@@ -6,6 +6,10 @@
 
 For every positive integer ``n`` with ``gcd(n, 10) = 1``,
 there exists a repunit ``R_k`` divisible by ``n``.
+
+References:
+- https://en.wikipedia.org/wiki/Repunit
+- https://en.wikipedia.org/wiki/Multiplicative_order
 """
 
 from math import gcd