Hi Guys,
Need a little help with this one, Not sure how to proceed.
Two positive integers, are connected by the relation . By using a binomial expansion show that for all positive integral values of n the expression is exactly divisible by . By choosing suitable values of a and n, show that is divisible by 9, and hence that is divisible by 9.
I tried putting the substitution for into the equation, and did a binomial expansion of , but not sure where that is supposed to lead. It seems like another longer equation.
Thanks for your help.
Hello,
Some steps have been volutarily skipped.
let's expand
Considering it mod bē, we can remove the powers of b superior to 2, because if for example , b^2 can be factored out and since b^2=0 mod b^2, it cancels all the terms with k>2.
Hence
and it gives your result
Thanks guys. @Moo, sorry to sound stupid. I got the summation part, but the rest of the notation went way over my head.
@melese. Got it! The 1 and 2nb terms cancel out, rest all terms have powers of b >= 2. Then rest a=2,b=3 and n=20, forms similar expression hence divisible, which checks out. Thanks!