SOLUTION: Determine all integers k such that {{{k^3+k+1}}} is divisible by 11.

Algebra ->  Divisibility and Prime Numbers -> SOLUTION: Determine all integers k such that {{{k^3+k+1}}} is divisible by 11.      Log On


   

Question 1019201: Determine all integers k such that k%5E3%2Bk%2B1 is divisible by 11.
Answer by Edwin McCravy(20064) About Me  (Show Source):
You can put this solution on YOUR website!
k%5E3%2Bk%2B1

We see that when k=2

2%5E3%2B2%2B1=8%2B2%2B1=11

And if we add any multiple of 11 to 2,

%282%2B11n%29%5E3%2B%282%2B11n%29%2B1

We get

2%5E3%2B3%282%29%5E2%2811n%29%2B3%282%29%2811n%29%5E2%2B%2811%29%5E3%2B2%2B11n%2B1

which is 2%5E3%2B2%2B1=11 plus some multiples of 11,

so integers of the form 2+11n, when n is a non-
negative integer, substituted for k into k%5E3%2Bk%2B1 
will always yield a number divisible by 11.

They are the positive integers 2 mod 11.

Edwin