SOLUTION: I must find, by a method other than induction, that 1+2^(6K+1)is divisible by 3.

Question 699138: I must find, by a method other than induction, that 1+2^(6K+1)is divisible by 3.
Answer by KMST(5328) (Show Source): You can put this solution on YOUR website!
I can prove that even more similarly defined numbers are divisible by 3.
If you (or the teacher do not like a generalization, you can always adapt the proof to just the subset of numbers.

All numbers of the form are divisible by 3.
Some of them (but not all) are of the form , if ,
as for (with , or )
.

so .
+ ... + [ + ... + ]+ , calling the messy bracket .
The conclusion so far is (in words) the even powers of 2, when divided by 3, have 1 as the remainder.
In formulas: , where is a whole number.

Since is times the whole number ,
is divisible by .
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