1950 AHSME Problems/Problem 20

Problem

When $x^{13}+1$ is divided by $x-1$ , the remainder is:

$\textbf{(A)}\ 1\qquad\textbf{(B)}\ -1\qquad\textbf{(C)}\ 0\qquad\textbf{(D)}\ 2\qquad\textbf{(E)}\ \text{None of these answers}$

Solution

Solution 1

Use synthetic division. Notice that no matter what the degree of $x$ of the dividend is, the remainder is always $\boxed{\mathrm{(C)}\ 0.}$

Solution 2

Notice that $1$ is a zero of $x^{13} - 1$ . By the factor theorem, since $1$ is a zero, then $x-1$ is a factor of $x^{13} - 1$ , and when something is divided by a factor, the remainder is $\textbf{(C)}0$

1950 AHSC (Problems • Answer Key • Resources)
Preceded by Problem 19	Followed by Problem 21
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1950 AHSME Problems/Problem 20

Contents

Problem

Solution

Solution 1

Solution 2

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