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zmudz
 one year ago
Compute the number of ordered pairs of integers \((x,y)\) with \(1\le x<y\le 100\) such that \(i^x+i^y\) is a real number.
zmudz
 one year ago
Compute the number of ordered pairs of integers \((x,y)\) with \(1\le x<y\le 100\) such that \(i^x+i^y\) is a real number.

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anonymous
 one year ago
Best ResponseYou've already chosen the best response.0\[x=\left\{ 2m,m \in N,1\le m \le 49 \right\}\] \[y=\left\{ 2n,n \in N,2\le n \le 50 \right\}\] \[\left( x,y \right)=\left\{( 2m,2n),m<n,m \in N,n \in N,1\le m \le 49,2\le n \le 50 \right\}\]

thomas5267
 one year ago
Best ResponseYou've already chosen the best response.1But x=1 and y=3 would also work right?

zmudz
 one year ago
Best ResponseYou've already chosen the best response.0@surjithayer @thomas5267 , I got (1, 3), (3, 1), (0, 2), and (2, 0) but 4 ordered pairs is wrong :( I can't get the answer unless I give up on the problem, but I really need the points. I'm not sure what @surjithayer 's notation in the set means. Any further help is greatly appreciated!

thomas5267
 one year ago
Best ResponseYou've already chosen the best response.1\(0\leq x\color{red}{<}y\leq100\)

thomas5267
 one year ago
Best ResponseYou've already chosen the best response.1\(i\) is the imaginary number right?

anonymous
 one year ago
Best ResponseYou've already chosen the best response.0x and y are even numbers when m=1,x=2*1=2 when n=2 y=2*2=4 \[\iota ^2=1\]

thomas5267
 one year ago
Best ResponseYou've already chosen the best response.1Nope, x and y being even is not the only cases when \(i^x+i^y\) is real. \(i+i=0\in\mathbb{R},\,1+1=2\in\mathbb{R}\)

thomas5267
 one year ago
Best ResponseYou've already chosen the best response.1\[ \begin{align*} x&=2n,\,1\leq n\leq49\\ y&=2m,\,2\leq n+1\leq m \leq 50\\ n&=1,\,x=2,\,y=4,6,8,\dots,100\\ n&=1,\,x=2,\,49\text{ choices of }y\\ n&=2,\,x=4,\,y=6,8,\dots,100\\ n&=2,\,x=4,\,48\text{ choices of }y \end{align*} \] Choices of \((x,y)\) with \(x=2n,\,1\leq n\leq49,\,y=2m,\,2\leq n+1\leq m \leq 50\): \(49+48+47+\cdots+3+2+1=50\times49\div 2=1225\) \[ \begin{align*} x&=2n1,\,1\leq n\leq49\\ y&=2m1,\,2\leq n+1\leq m \leq 50\\ n&=1,\,x=1,\,y=3,5,7,\dots,99\\ n&=1,\,x=1,\,49\text{ choices of }y\\ n&=2,\,x=3,\,y=5,7,9,\dots,99\\ n&=2,\,x=3,\,48\text{ choices of }y \end{align*} \] Choices of \((x,y)\) with \(x=2n1,\,1\leq n\leq49,\,y=2m1,\,2\leq n+1\leq m \leq 50\): \(49+48+47+\cdots+3+2+1=50\times49\div 2=1225\) \[ \text{Total} = 1225\times2=2450 \]

thomas5267
 one year ago
Best ResponseYou've already chosen the best response.1Something is wrong with my answer.

thomas5267
 one year ago
Best ResponseYou've already chosen the best response.1\[ \begin{align*} x&=4n3,\,1\leq n\leq 25\\ y&=4m1,\,1\leq n\leq m \leq 25\\ n&=1,\,x=1,\,y=3,7,11,\dots,99\\ n&=1,\,x=1,\,25\text{ choices of }y\\ n&=2,\,x=5,\,y=7,11,15,\dots,99\\ n&=2,\,x=3,\,24\text{ choices of }y \end{align*} \] Choices of \((x,y)\) with \(x=4n3,\,1\leq n\leq 25,\,y=4m1,\,1\leq n\leq m \leq 25\): \(25+24+23+22+\cdots+2+1=26\times25\div2= 325\) \[ \begin{align*} x&=4n1,\,1\leq n\leq 24\\ y&=4m3,\,2\leq n+1\leq m \leq 25\\ n&=1,\,x=3,\,y=5,9,13,\dots,97\\ n&=1,\,x=3,\,24\text{ choices of }y\\ n&=2,\,x=7,\,y=9,13,17,\dots,97\\ n&=2,\,x=3,\,23\text{ choices of }y \end{align*} \] Choices of \((x,y)\) with \(x=4n1,\,1\leq n\leq 24,\,y=4m3,\,2\leq n+1\leq m \leq 25\): \(24+23+22+21+\cdots+2+1=25\times24\div2= 300\) Total=\(300+325+1225=1850\) This is the answer returned by Mathematica.

zmudz
 one year ago
Best ResponseYou've already chosen the best response.01850 is the right answer. The teacher got it by doing casework: (a) \(i^x = i\) and \(i^y = i\), or (b) \(i^x = i\) and \(i^y = i\) They both give 625 cases, because there are 25 numbers n for which \(i^n = i\) (namely, \(n = 1, 4, \ldots, 97\)), and there are 25 numbers n for which \(i^n = i\) (namely \(n = 3, 7, \ldots, 99\)). This gives 1250 cases total. There are 2500 other cases (50*50) that result from having any pair of even numbers in which it will give a real answer. But then there are overlaps of 50 cases when x=y, so you subtract that. And by symmetry, only half of the remaining will satisfy x<y, so that gives 1850.

thomas5267
 one year ago
Best ResponseYou've already chosen the best response.1Basically the same idea but your teacher count all combinations then subtract duplicates where I count all possible combinations directly at a cost of a higher complexity.
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