From db2ac78ab8cc36daf2f5ba6c6e76ad7234efc900 Mon Sep 17 00:00:00 2001
From: Felix Martin <mail@felixm.de>
Date: Sun, 16 Aug 2026 09:53:53 -0400
Subject: [PATCH] Solve problem 274

---
 python/e274.py | 57 ++++++++++++++++++++++++++++++++++++++++++++++++++
 1 file changed, 57 insertions(+)
 create mode 100644 python/e274.py

diff --git a/python/e274.py b/python/e274.py
new file mode 100644
index 0000000..bc1b81b
--- /dev/null
+++ b/python/e274.py
@@ -0,0 +1,57 @@
+from math import gcd
+from lib_prime import primes
+from lib_misc import modinv
+
+
+def find_m(p: int) -> int:
+    rs = []
+    test_n = p * 1337
+    for m in range(1, p):
+        test_f = f(test_n, m)
+        if test_f % p == 0:
+            rs.append(m)
+    assert len(rs) == 1
+    return rs[0]
+
+
+def calc_m(p: int) -> int:
+    """
+    (n // 10 + (n % 10) * m) % p == 0
+    a + b * m == 0 (mod p)
+        b * m == -a (mod p)
+            m == -a * b**-1 (mod p)
+    """
+    n = p * 1337
+    a = n // 10
+    b = (n % 10)
+    m = -a * modinv(b, p)
+    m %= p
+    return m
+
+
+def f(n: int, m: int = 34) -> int:
+    left = n // 10
+    right = n % 10 * m
+    r = left + right
+    return r
+
+
+def euler_274():
+    ps = primes(10**7)
+    sum_m = 0
+    for p in ps:
+        if gcd(p, 10) != 1:
+            continue
+        m = calc_m(p)
+        if p < 1000:
+            m_naiv = find_m(p)
+            assert m == m_naiv
+        sum_m += m
+    return sum_m
+
+
+if __name__ == "__main__":
+    solution = euler_274()
+    print(f"e274.py: {solution}")
+    assert solution == 1601912348822
+