jfranciscocuencag wrote:
Hi...
When you divide any number by 10, the remainder is the units digit..
WHY?..
Because the closest multiple of 10 to that number will be the last number having units digit as 0. So whatever extra is there in units digit is the remainder.
So we have to find the units digit of the term.
All digits have a cylicity when it comes to units digit...
3 gives 3,9,7,1,3,9,7,1,3,9.....
\(3^1=3;3^2=9;3^3=27,3^4=81;3^5=243\)..unit digits are 3,9,7,1,3...
Also any number having 3 as units digit will always have same cylicity as 3, so let it be 1876543 or just 3, both will follow same cylicity...
Therefore \(13^{4k+2}\) will have same units digit as 13^2, so units digit will be 9..
So the units digit of entire term will be 9+8=17, thus 7 will be the remainder.
A
Thank you for your answer!
I just have one question, how do we know that the cyclicity number that we have to choose is 9?
Kind regards![/quote]
Hi ..
We know cylicity is 3,9,7,1,3,9,...
So the cylicity is in 4s...(3,9,7,1),(3,9,7,1)....
So the cylicity is same for first, fifth, ninth terms.... And for second, sixth, tenth and so on.
That is every fourth term is same...
Here we have 13^{4k+2}...
Now 4k means every fourth term, that is 13^{4k} will have _,_,_,1,_,_,_,1...that is it will have 1..
But we are looking for (4k+2)th term and it will have same number as the 2nd term, and, therefore, it is 9...
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