Not quite sure where this belongs...

[Guest]

Hello, all. I'm not sure how to write a formula based on a conjecture (I'm in pre-calculus, for Pete's sake --- I'm doing this just for fun on my part), but I would like to try and write one based on the following hypothesis:

Any number with the final digit of six with any exponent will always have a solution with the final digit also being six.



1) Does that make any sense?
2) What kind of equation can I write so it would be in terms of, say, "Prove [suchandsuch] so that it equals [soandso]."


Any help would be greatly appreciated.

 

[daon]

Hello, all. I'm not sure how to write a formula based on a conjecture (I'm in pre-calculus, for Pete's sake --- I'm doing this just for fun on my part), but I would like to try and write one based on the following hypothesis:

Any number with the final digit of six with any exponent will always have a solution with the final digit also being six.



1) Does that make any sense?
2) What kind of equation can I write so it would be in terms of, say, "Prove [suchandsuch] so that it equals [soandso]."


Any help would be greatly appreciated.

I think you want to prove:

Let n be an integer, a be a positive integer.
For n having a six in the one's place, n^a will have a solution with a six in the one's place.

You will want to show that:

Given \(\displaystyle \L n = \sum_{i=0}^{k-1}x_i 10^i\) where \(\displaystyle x_0 10^0 = 6\) that

\(\displaystyle \L [\sum_{i=0}^{k-1}x_i 10^i]^a = \sum_{j=0}^{l-1}y_j 10^j \Leftrightarrow y_0 10^0 = 6\)

Now I don't know your level of skill, but when I was in precalculus I wouldn't even know how to attempt this. Note that the two sums represent two different numbers, one is your original number and the next is the solution (k is the number of digits in n, l is the number of digits in the solution).. and the scary sums you see up there are only a succession of additions like:

\(\displaystyle 1625 = 5 (10^0) + 2(10^1) + 6(10^2) + 1(10^3)\)

Please also note that there may be an easier way to do this, as I am fairly new to proofs myself..

 

[Guest]

Ooh, neat. I was just playing with numbers today and I was thinking about it. We'll see if anyone else has anymore input, yeah? Thanks muchly.

 

[Guest]

P.S. This is what I wrote down initially...

(_6)^n = _6


I don't know how to do the fancy text or write that in a more mathematical way, sorry.

 

[daon]

P.S. This is what I wrote down initially...

(_6)^n = _6


I don't know how to do the fancy text or write that in a more mathematical way, sorry.

Your intuition is good, and that is very important in higher-maths. You will learn soon enough how to do proofs (via a classroom), but it is excellent you are challenging yourself so early-on. I know member Pka is good with this kind of thing.