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Difference between revisions of "1980 AHSME Problems/Problem 17"

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-aopspandy, edited by j314andrews
 
-aopspandy, edited by j314andrews
  
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== Solution ==
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== Solution 2 ==
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Since we have an imaginary term, we can think about rotations. We are in the first and second quadrant, so we only need to think about angles from 0 to <math>\pi</math> exclusive. Specifically, <math>4\theta = \pi k</math>, where <math>k</math> is an integer. Therefore, the only angles which can work are <math>\pi/4, \pi/2</math> and <math>3\pi/4</math>.
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For <math>(n+i)^4</math> to be a real number, <math>(n+i)^4</math> must be a scalar multiple of an eighth root of unity, that is, <math>n+i = r\left(\cos \frac{k\pi}{4} + i \sin \frac{k\pi}{4}\right)</math> where <math>r \geq 0</math> is a real number and <math>k</math> is an integer such that <math>0 \leq k \leq 7</math>. So <math>r \sin \frac{k\pi}{4} = 1</math> and <math>\sin \frac{k\pi}{4}</math> is positive.  Therefore, <math>k \in \{1, 2, 3\}</math>.  
  
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Now we just need to see if these angles can be represented by <math>(n+i)^4</math>. <math>\pi/4</math> and <math>3\pi/4</math> work, since they form a 45-45-90 triangle, and <math>\pi/2</math> works, since it doesn't have a real component.
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If <math>k = 1</math>, <math>r = \frac{1}{\sin\frac{\pi}{4}} = \sqrt{2}</math>, <math>n = \sqrt{2}\cos\frac{\pi}{4} = 1</math> and <math>(n+i)^4 = (1+i)^4 = -4</math>.
  
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So, the answer is <math>\boxed{D}</math>.
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If <math>k = 2</math>, <math>r = \frac{1}{\sin\frac{\pi}{2}} = 1</math>, <math>n = \cos\frac{\pi}{2} = 0</math> and <math>(n+i)^4 = i^4 = 1</math>.
  
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~ jaspersun
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If <math>k = 3</math>, <math>r = \frac{1}{\sin\frac{3\pi}{4}} = \sqrt{2}</math>, <math>n = \sqrt{2}\cos\frac{3\pi}{4} = -1</math> and <math>(n+i)^4 = (-1+i)^4 = -4</math>. 
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Therefore, <math>n \in \{1, 0, -1\}</math>, so there are <math>\boxed{(\mathbf{D})\ 3}</math> possible values of <math>n</math>.
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~ jaspersun, edited by j314andrews
  
 
== See also ==
 
== See also ==

Revision as of 13:04, 15 August 2025

Problem

Given that $i^2=-1$, for how many integers $n$ is $(n+i)^4$ an integer?

$\text{(A)} \ \text{none} \qquad \text{(B)} \ 1 \qquad \text{(C)} \ 2 \qquad \text{(D)} \ 3 \qquad \text{(E)} \ 4$


Solution 1

Expanding $(n+i)^4$ yields $n^4+4in^3-6n^2-4in+1 = (n^4-6n^2+1) + (4n^3-4n)i$. This quantity is an integer if and only if $4n^3 - 4n = 4n(n+1)(n-1) = 0$, that is, if $n \in \{0, 1, -1\}$. Therefore, there are $\boxed{(\textbf{D})\ 3}$ such values of $n$.

-aopspandy, edited by j314andrews

Solution 2

For $(n+i)^4$ to be a real number, $(n+i)^4$ must be a scalar multiple of an eighth root of unity, that is, $n+i = r\left(\cos \frac{k\pi}{4} + i \sin \frac{k\pi}{4}\right)$ where $r \geq 0$ is a real number and $k$ is an integer such that $0 \leq k \leq 7$. So $r \sin \frac{k\pi}{4} = 1$ and $\sin \frac{k\pi}{4}$ is positive. Therefore, $k \in \{1, 2, 3\}$.

If $k = 1$, $r = \frac{1}{\sin\frac{\pi}{4}} = \sqrt{2}$, $n = \sqrt{2}\cos\frac{\pi}{4} = 1$ and $(n+i)^4 = (1+i)^4 = -4$.

If $k = 2$, $r = \frac{1}{\sin\frac{\pi}{2}} = 1$, $n = \cos\frac{\pi}{2} = 0$ and $(n+i)^4 = i^4 = 1$.

If $k = 3$, $r = \frac{1}{\sin\frac{3\pi}{4}} = \sqrt{2}$, $n = \sqrt{2}\cos\frac{3\pi}{4} = -1$ and $(n+i)^4 = (-1+i)^4 = -4$.

Therefore, $n \in \{1, 0, -1\}$, so there are $\boxed{(\mathbf{D})\ 3}$ possible values of $n$.

~ jaspersun, edited by j314andrews

See also

1980 AHSME (Problems • Answer Key • Resources)
Preceded by
Problem 16 		Followed by
Problem 18
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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