11) Let $P(x)$ be a polynomial whose coefficients are either 0 or 1. State which of the following statements is true
(A) $P(2)$ cannot be 51
(C) $P(3)$ cannot be 92
(B) $P(3)$ cannot be 37
(D) $P(4)$ cannot be 20
(E) $P(5)$ cannot be 150.
11) The answer is (C). We observe that if $P(x)=a_{n} x^{n}+a_{n-1} x^{n-1}+\ldots+a_{0}$ is a polynomial with coefficients 0 or 1, then for every integer $k$ the number $P(k)$ is represented in base $k$ by the digits $\left(a_{n} a_{n-1} \ldots a_{0}\right)_{k}$. Since the base 3 representation of $(92)_{10}$ is (10102) $)_{3}$, and thus contains the digit 2, it is impossible to have $P(3)=92$. On the other hand, (A) is false by taking $p(x)=x^{5}+x^{4}+x+1$, (B) is false by taking $p(x)=x^{3}+x^{2}+1$, (D) is false by taking $p(x)=x^{2}+x$, (E) is false by taking $p(x)=x^{3}+x^{2}$.

SECOND SOLUTION

A polynomial $p(x)$ with coefficients 0 or 1, when divided by $x$, must give a remainder of 0 or 1 (the constant term). Therefore, (C) is true since 92:3 gives a remainder of 2.