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One day Bob and Alice walk around they were having the following conversation:


Bob: Alice go ahead and pick any random number.

Alice: Okay, I have one.

Bob: Alice multiply the random number by X and sum every single digit(s) until it's Y digit(s) left.

Alice: Done.

Bob: I would bet that the result is an odd number.

Alice: Well, yeah. It's either odd or even either way. You just got lucky!

Bob: How about I said that it was X?

Alice: How do you know?

Bob: Well, that's simply because <reasoning>


What number are X and Y? Can anyone explain Bob's <reasoning>?

Fun fact: There is a related thread in Math.SE which explains Bob's reasoning. When the question digits are added it would be X-4.

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X is 9 and Y is 1, because whenever you multiply a number by nine, the sum of its digits is either nine or a multiple of nine. Nine is the only one digit number that is a multiple of nine.

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Take an odd Y-digit number which equals to $X$. Let the original number be $O$. If we go through all the steps backwards, a bunch of odd or even numbers can be obtained. If $X=9n+m$, $O(9n+m) = 9k+m$, which means $m$ can only be 0 if it is to always work regardless of $O$. If even 2 as $2X$ ends up as $X$, in this case there can be only one step, and $X=9$, $Y=1$.

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