Discrete math induction proof (divisibilty). How to show that 10^n-(-1)^n is always divisible by 11 through proof of induction?

curukksm

curukksm

Answered question

2022-09-07

Discrete math induction proof (divisibilty)
How to show that 10 n ( 1 ) n is always divisible by 11 through proof of induction?

Answer & Explanation

Lena Ibarra

Lena Ibarra

Beginner2022-09-08Added 13 answers

Step 1
10 n ( 1 ) n = ( 11 ) i = 0 n 1 10 i
But, if this is an exercise in writing proofs by induction.
base case: n = 1 10 1 ( 1 ) 1 = 11
Step 2
Inductive hypothesis: Assume that 10 n ( 1 ) n is divisible by 11.
show that: 10 n + 1 ( 1 ) n + 1 is divisible by 11.
10 ( 10 n ) + ( 1 ) n 10 ( 10 n ) 10 ( 1 ) n + 11 ( 1 ) n 10 ( 10 n ( 1 ) n ) + 11 ( 1 ) n
11 divides the fist term from the inductive hypothesis, and clearly 11 divides the 2nd term.
Nodussimj

Nodussimj

Beginner2022-09-09Added 14 answers

Step 1
An obvious generalization:
If b N and b 2 then ( b k ) n ( k ) n is always divisible by b for n , k N .
Step 2
Proof (one of many): Since u n v n = ( u v ) j = 0 n 1 u j v n 1 j ,
( b k ) n ( k ) n = ( ( b k ) ( k ) ) j = 0 n 1 ( b k ) j ( k ) n 1 j = ( b ) j = 0 n 1 ( b k ) j ( k ) n 1 j
and this is always divisible by b.

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