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# Let m be a positive integer

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Let m be a positive integer, and suppose that 9 is its own inverse mod m, but 2 is not its own inverse . How many possible values for m are there?

Nov 25, 2020

This would mean that $9m \equiv 1 \pmod m.$ Therefore, $0 \equiv 1 \pmod m,$ so $m = 1.$ But, if we plug this in to $2m \equiv 1 \pmod m,$ we get 2 = 1 mod 1 which is obviously true. Therefore, there is no solution.