Mock AIME I 2015 Problems/Problem 8
Let be consecutive terms (in that order) in an arithmetic sequence with common difference . Suppose and are roots of a monic quadratic with . Then for positive relatively prime integers and . Find the remainder when is divided by .
Let and and substitute, then use the trigonometric identities and to find that
Furthermore, we have that for some numbers and and that . We also know that the roots of are and , so it follows by Vieta's formulas that and that . Hence
and We see that and are already relatively prime; hence,