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[2019-Jul] Fun modulo 5

Could you believe it? Stefan was playing around with his favorite numbers, he wanted to practice taking the fifth power of them, when suddenly he realized, that the least significant digit of the result was the same as the original number's. Can you show him why?

That is, prove: n mod 10 = (n ^ 5) mod 10

Resources

Download Files

Definitions File 

theory Defs
  imports Main
begin
end

Template File 

theory Submission
  imports Defs
begin

lemma modpower5: fixes n :: nat
  shows "n mod 10 = (n ^ 5) mod 10" sorry


end

Check File 

theory Check
imports Submission
begin
                     
lemma "(n::nat) mod 10 = (n ^ 5) mod 10"
by (rule Submission.modpower5)

end
Download Files

Definitions File 

Require Export Arith Lia.

Template File 

Require Import Defs.

(* Proving this might be useful (but is not mandatory). *)
Lemma mod_power a b n :
  ((a mod b) ^ n) mod b = (a ^ n) mod b.
Admitted.

Theorem modpower5 : forall (n: nat),
  n mod 10 = (n ^ 5) mod 10.
Proof.
  (* todo *)
Admitted.
Download Files

Definitions File 

theory Defs
  imports Main
begin
end

Template File 

theory Submission
  imports Defs
begin

lemma modpower5: fixes n :: nat
  shows "n mod 10 = (n ^ 5) mod 10" sorry


end

Check File 

theory Check
imports Submission
begin
                     
lemma "(n::nat) mod 10 = (n ^ 5) mod 10"
by (rule Submission.modpower5)

end
Download Files

Definitions File 

-- no definitions required
-- Lean version: 3.4.2
-- Mathlib version: 2019-07-31

Template File 

theorem mod_power_five : ∀ (n : ℕ), n % 10 = (n ^ 5) % 10 :=
sorry

Check File 

import .submission

theorem you_did_it : ∀ (n : ℕ), n % 10 = (n ^ 5) % 10 := mod_power_five
Download Files

Definitions File 

(in-package "ACL2")

Template File 

(in-package "ACL2")

(defthm modpower5
  (implies (natp n)
           (equal (mod-expt n 5 10)
                  (mod n 10)))
                  )

Check File 

; The four lines just below are boilerplate, that is, the same for every
; problem.

(in-package "ACL2")
(include-book "Submission")
(set-enforce-redundancy t)
(include-book "Defs")

; The events below represent the theorem to be proved, and are copied from
; template.lisp.

(defthm modpower5
  (implies (natp n)
           (equal (mod-expt n 5 10)
                  (mod n 10)))
                  )
Download Files

Definitions File 

-- no definitions required
-- Lean version: 3.16.2
-- Mathlib version: eb5b7fb7f406385cd1f2efaa15d9c0923541b955

Template File 

theorem mod_power_five : ∀ (n : ℕ), n % 10 = (n ^ 5) % 10 :=
sorry

Check File 

import .submission

theorem you_did_it : ∀ (n : ℕ), n % 10 = (n ^ 5) % 10 := mod_power_five
Download Files

Definitions File 

theory Defs
  imports Main
begin
end

Template File 

theory Submission
  imports Defs
begin

lemma modpower5: fixes n :: nat
  shows "n mod 10 = (n ^ 5) mod 10" sorry


end

Check File 

theory Check
imports Submission
begin
                     
lemma "(n::nat) mod 10 = (n ^ 5) mod 10"
by (rule Submission.modpower5)

end
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