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October 3rd, 2015, 02:46 PM   #1
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Impossible proof by induction

Prove (3^k ) + (7 ^ (k-1)) + 8 is divisible by 12.

Last edited by skipjack; October 3rd, 2015 at 03:08 PM.
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October 3rd, 2015, 02:50 PM   #2
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Why is this impossible?
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October 3rd, 2015, 04:26 PM   #3
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True for k=1.
Assume true for k, show true for k+1
(for k+1) $\displaystyle 3(3^k)+7(7^{k-1})+8$ divisible by 12?
Subtract assumption for k and get $\displaystyle 2(3^k)+6(7^{k-1})$ divisible by 12?
Expression is $\displaystyle 6(3^{k-1}+7^{k-1})$
Term inside parentheses is even! Therefore 6x term is divisible by 12!
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