1. A math competition consists of 90 multiple-choice questions. Camilla answered all the questions: which of the following cannot be Camilla's total score, knowing that a correct answer is worth 5 points and a wrong answer is worth -1 point?
(A) -78
(B) 116
(C) 204
(D) 318
(E) 402
1. The answer is (B). Let $r$ be the number of correct answers given by Camilla; the number of wrong answers is $90-r$, and the total score is $5 r+(-1)(90-r)=6 r-90=6(r-15)$. It follows that Camilla's score is a multiple of 6, and therefore cannot be equal to $116=$ $2^{2} \cdot 29$, since this number is not divisible by 3. All other scores are possible: Camilla can score $-78, 204, 318$, and 402 points by giving 2, 49, 68, and 82 correct answers, respectively; indeed, Camilla's score can be any multiple of 6 between $6 \cdot(0-15)=-90$ and $6 \cdot(90-15)=450$.