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"Nathan claims that if you pick a two-digit number whose units digit is odd, but not 5, such as 37, and multiply it by some positive integer n and tell him the last two digits of your result that he can tell you the remainder when n is divided by 25. Is Nathan's claim possible? Why or why not?"

So, the units digit is odd and not $5$. That means $100$ and our number is coprime. I know that $100 = 2^25^2$, but from here I don't know what other conclusions I can make.

Help?

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    $\begingroup$ If you tell him 66, how is he suppose to know if n is 6 and your original number is 11) or 2 (and your number is 33)? $\endgroup$ – T.J. Gaffney Dec 23 '13 at 21:16
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    $\begingroup$ Does Nathan know the two digit number which is relatively prime to $10$? $\endgroup$ – robjohn Dec 23 '13 at 21:26
  • $\begingroup$ Nathan can claim that $n$ is some particular number, but $3$ and $1$ are equivalent modulo $10$ under multiplication by $\{4, 2\}$. What mechanism could Nathan use to separate these results? $\endgroup$ – abiessu Dec 23 '13 at 21:48
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Nathan's claim can be stated as follows: let $x$ be a number that is coprime to $100$, then given the equation $$ nx \equiv c \pmod{100} $$ Where $x$ and $c$ are known, we may determine the value of $n$ modulo $25$.

Hint: Chinese remainder theorem

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  • $\begingroup$ Sorry, but what would $c$ be since you said that $x$ and $c$ are known? So $n$ is the integer we are multiplying by, $x$ is the two digit integer that is coprime with $100$, and $c$ is then the remainder of $nx$ modulo $100$? $\endgroup$ – asd Dec 23 '13 at 21:24
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    $\begingroup$ I'm thinking of the problem from Nathan's perspective. $n$ is the secret number we're multiplying by $x$, which is coprime to $100$. $c$ is "the last two digits" of the product. $\endgroup$ – Omnomnomnom Dec 23 '13 at 21:34
  • $\begingroup$ How do we know that $c$ is the last two digits of the product? $\endgroup$ – asd Dec 23 '13 at 22:14
  • $\begingroup$ Or rather, why are we able to create this equation $\endgroup$ – asd Dec 23 '13 at 22:21
  • $\begingroup$ Again, I'm thinking about the problem from nathan's perspective. I know the number (coprime to $100$) that was multiplied by $n$, and I've called this $x$. The other person has multiplied $x$ by his mystery number $n$ and told me the last two digits, and I'm calling this number (that I've been given) "$c$". The equation simply states that if you multiply $n$ by $x$ and extract the last two digits, you get $c$. Does that make sense? $\endgroup$ – Omnomnomnom Dec 24 '13 at 23:55

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