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I am studying maths and have the following question set by my professor.

I thought the answer was 5, because: 3 . 5 = 15, and 15 mod 7 is 1. Since 1 mod 7 = 1 that means they match.

However, I looked at the formula and noticed both variables are z, which leads me to believe they both have to be 3. Look for the red in the picture. enter image description here I guess my question is what the answer is, and why?

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  • $\begingroup$ You may let $z^{-1} = x$ and proceed. $z^{-1}$ is different, it is not $\dfrac{1}{z}$. $\endgroup$ – AgentS Jul 28 '19 at 23:36
  • $\begingroup$ @plagiarism The $z$ values don't have to be the same (and, in this case, are not) in each question. They're just reusing a variable letter. $\endgroup$ – John Omielan Jul 28 '19 at 23:39
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    $\begingroup$ @rsadhvika $z^{-1}=x$ is a bad idea, because $x$ is used for the modulus in the first equation. $\endgroup$ – Andreas Blass Jul 28 '19 at 23:47
  • $\begingroup$ Ah I didn't notice that! bad idea agree :) Ty @a $\endgroup$ – AgentS Jul 28 '19 at 23:52
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    $\begingroup$ The question is so badly worded that I find it hard to believe it came from a maths professor. $\endgroup$ – TonyK Jul 29 '19 at 0:05
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The first line, where you have underlined it in red, is the definition of $z^{-1}$. The next two lines are problems you are to do. Your answer of $5$ for the first is correct.

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$3^{-1}\cong 5\pmod 7$ and $2^{-1}\cong3\pmod 5$. The first is because $3\cdot 5=15\cong1\pmod7$. The second because $2\cdot3=6\cong1\pmod5$.

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  • $\begingroup$ If my answer to the question is correct (I said it was 5) I do not understand why my professor had two variables with the same letter z, when clearly they hold different numbers?! :'-( $\endgroup$ – plagiarism Jul 28 '19 at 23:44
  • $\begingroup$ @plagiarism Mira and Mira's husband are two different people. Do you see that? $\endgroup$ – AgentS Jul 28 '19 at 23:45
  • $\begingroup$ Two different problems. $\endgroup$ – Chris Custer Jul 28 '19 at 23:46
  • $\begingroup$ Okay. So just to double check, my answer of 5 was correct, and z and z^-1 are two different numbers? $\endgroup$ – plagiarism Jul 28 '19 at 23:48
  • $\begingroup$ Yes. $z$ and $z^{-1}$ are different; and there are two problems. $\endgroup$ – Chris Custer Jul 28 '19 at 23:49
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The professor gives the general form as

$$z \cdot z^{-1} \equiv 1 \pmod x$$

where $x,z,z^{-1}$ are integers.

For the first question,

$$3\cdot z^{-1} \equiv 1 \pmod 7$$

setting

$$z^{-1}=\{5,12,19,26,\dots\}$$

will work. For the second question,

$$2\cdot z^{-1} \equiv 1 \pmod 5$$

setting

$$z^{-1}=\{3,8,13,18,\dots\}$$

will work. There are multiple values to assign for $z^{-1}$ depending on the context.

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The answer is that the z variables are different numbers, because one z is to the power of negative one.

The answer 5 was also correct for the example question.

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