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Difference between revisions of "1950 AHSME Problems/Problem 20"

(Created page with "== Problem== When <math>x^{13}-1</math> is divided by <math>x-1</math>, the remainder is: <math> \textbf{(A)}\ 1\qquad\textbf{(B)}\ -1\qquad\textbf{(C)}\ 0\qquad\textbf{(D)}\ 2...")
 
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<math> \textbf{(A)}\ 1\qquad\textbf{(B)}\ -1\qquad\textbf{(C)}\ 0\qquad\textbf{(D)}\ 2\qquad\textbf{(E)}\ \text{None of these answers} </math>
 
<math> \textbf{(A)}\ 1\qquad\textbf{(B)}\ -1\qquad\textbf{(C)}\ 0\qquad\textbf{(D)}\ 2\qquad\textbf{(E)}\ \text{None of these answers} </math>
  
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==Solution==
+
==Solution 1==
  
 
Use synthetic division. Notice that no matter what the degree of <math>x</math> of the dividend is, the remainder is always <math>\boxed{\mathrm{(C)}\ 0.}</math>
 
Use synthetic division. Notice that no matter what the degree of <math>x</math> of the dividend is, the remainder is always <math>\boxed{\mathrm{(C)}\ 0.}</math>
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 +
==Solution 2==
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 +
Notice that <math>1</math> is a zero of <math>x^13 - 1</math>. By the factor theorem, since <math>1</math> is a zero, then <math>x-1</math> is a factor of <math>x^13 - 1</math>, and when something is divided by a factor, the remainder is <math>\textbf{(C)}0</math>
  
 
==See Also==
 
==See Also==
  
 
{{AHSME box|year=1950|num-b=19|num-a=21}}
 
{{AHSME box|year=1950|num-b=19|num-a=21}}

Revision as of 10:40, 29 December 2011

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 $\textbf{(C)}0$

See Also

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
All AHSME Problems and Solutions
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