SOLUTION: Prove by induction and through divisibility algorithm that 11^n - 6 is divisible by 5 for every positive integer n.

Question 1188706: Prove by induction and through divisibility algorithm that 11^n - 6 is divisible by 5 for every positive integer n.
Answer by ikleyn(52787) (Show Source): You can put this solution on YOUR website!
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Prove by induction and through divisibility algorithm that 11^n - 6 is divisible by 5 for every positive integer n.
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(1)  Base case n= 1.

     Then   = 11 - 6 = 5  is divisible by 5,  so the base of induction is established.



(2)  The induction step from n to (n+1).


     We assume that    is divisible by 6 for some integer positive index n.


     Then   =  =  =  + (66-6) =  + 60.


     The addend    is divisible by 5 due to the inductive assumption, and the term 60 is also divisible by 5.


     Thus the inductive step from n to (n+1) is complete.



(3)  Due to the principle of Mathematical induction, the statement is proved for all positive integer n.

Solved.

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Above was the proof by induction.

Below is more simple proof using the divisibility by 5 rule.

The number    has the last (the units) digit  1 (one).


When we subtract 6 from this number, we get the last digit 5 for the difference,

which means that this difference,  ,  is divisible by 5 without a remainder.

Solved (twice, by two different methods).

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