Update Thu Feb 15 23:12:03 UTC 2024

Math2001/Homework/hw4.lean

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+/- Copyright (c) Heather Macbeth, 2023.  All rights reserved. -/
+import Library.Basic
+import Library.Tactic.ModEq
+import AutograderLib
+
+math2001_init
+
+
+/-! # Homework 4
+
+Don't forget to compare with the text version,
+https://github.com/hrmacbeth/math2001/wiki/Homework-4,
+for clearer statements and any special instructions. -/
+
+
+@[autograded 4]
+theorem problem1 (h1 : a ≡ b [ZMOD n]) (h2 : b ≡ c [ZMOD n]) : a ≡ c [ZMOD n] := by
+  sorry
+
+@[autograded 4]
+theorem problem2 {t : ℤ} (h : t ≡ 2 [ZMOD 4]) :
+    3 * (t ^ 2 + t - 8) ≡ 3 * (2 ^ 2 + 2 - 8) [ZMOD 4] := by
+  sorry
+
+@[autograded 4]
+theorem problem3 {a : ℤ} (ha : a ≡ 3 [ZMOD 5]) :
+    a ^ 3 + 4 * a ^ 2 + 3 ≡ 1 [ZMOD 5] := by
+  sorry
+
+@[autograded 4]
+theorem problem4 : ∃ k : ℤ, k > 50 ∧ k ≡ 2 [ZMOD 5] ∧ 5 * k ≡ 6 [ZMOD 8] := by
+  sorry
+
+@[autograded 5]
+theorem problem5 {x : ℤ} : x ^ 5 ≡ x [ZMOD 5] := by
+  sorry
+
+@[autograded 4]
+theorem problem6 {n : ℤ} (h1 : 5 ∣ n) (h2 : 12 ∣ n) : 60 ∣ n := by
+  sorry