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## Homework Statement

In a parlour game, the 'magician' asks one of the participants to think of a three-digit number [tex]abc_{10}[/tex]. Then the magician asks the participant to add the five numbers [tex]acb_{10},/bac_{10},/bca_{10},/cab_{10},/cba_{10}[/tex], and reveal their sum. Supppose the sum was 3194. What was the [tex]abc_{10}[/tex] originally?

## Homework Equations

none known

## The Attempt at a Solution

right, I am not positive I'm doing the right thing, but I hope I am on the right track, just say where I need to go if I'm not, or correct me, or do whatever, thnx

[tex]acb_{10} + bac_{10} + bca_{10} + cab_{10} + cba_{10} = 3194[/tex]

expand that

[tex]100a + 10c + b + 100b + 10a + c + 100b + 10c + a + 100c + 10a + b + 100c + 10b + a = 3194[/tex]

[tex]122a + 212b + 221c = 3194[/tex]

now, from here I'm kinda stuck. I have a hunch I have to use modular arithmetic somewhere (only just roughly learned it so that why I kinda stuck on how to use it).

If I was to guess I'd do something like

[tex]a (mod/122) + b (mod/212) + c (mod/221) = 3194[/tex]

but even with that guess id be stuck.

I 'could' try do a lot of guesswork and go from there but I'm guessing there's a more elegant way. thnx

hope you can help