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 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 AHSME (Problems • Answer Key • Resources)
Preceded by Problem 19	Followed by Problem 21
1 • 2 • 3 • 4 • 5 • 6 • 7 • 8 • 9 • 10 • 11 • 12 • 13 • 14 • 15 • 16 • 17 • 18 • 19 • 20 • 21 • 22 • 23 • 24 • 25 • 26 • 27 • 28 • 29 • 30
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1950 AHSME Problems/Problem 20

Contents

Problem

Solution 1

Solution 2

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