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# urgent! modular arithmetic watch

1. find all integer x in the range which satisfy

i can find all the x's just by plugging in the values in the range of 1 to 100, but i wanna know how to do this using the modular arithmetic.
thanks!
2. (Original post by singedang2)
find all integer x in the range which satisfy

i can find all the x's just by plugging in the values in the range of 1 to 100, but i wanna know how to do this using the modular arithmetic.
thanks!
If -2 is a quadratic residue there will be a solution in 0<x<9. Then -x is also a solution. And all other solutions will be of the form x+17k, 17k-x.
3. (Original post by RichE)
If -2 is a quadratic residue there will be a solution in 0<x<9.
i don't understand what this means...
(Original post by RichE)
Then -x is also a solution.
and how does the one above implies this?
(Original post by RichE)
And all other solutions will be of the form x+17k, 17k-x.
and how did you come up with this equation?
4. (Original post by singedang2)
and how does the one above implies this?
x^2 = (-x)^2.
5. (Original post by singedang2)
i don't understand what this means...

and how does the one above implies this?
what GB said.

and how did you come up with this equation?
because addition and multiplication are well-defined in modular arithmetic.

Sorry, no more help tonight, having run out of metaphorical toilet paper
6. (Original post by singedang2)
i don't understand what this means...
I don't either. I tried the good old-fashioned method of googling but found nothing useful. I have a feeling that 17/2 = 8.5 (rounded up to 9), as well as (-x)^2 = x^2, has something to do with it.

(Original post by singedang2)
and how did you come up with this equation?
If x is a solution, then x + 17 is a solution, x + 2*17 is a solution, x + 3*17 is a solution... same with -x.
7. (Original post by generalebriety)
If x is a solution, then x + 17 is a solution, x + 2*17 is a solution, x + 3*17 is a solution... same with -x.
i understand that. but i'm really interested in how that equation came up.
8. (Original post by singedang2)
i understand that. but i'm really interested in how that equation came up.
what equation?

x+17k = x mod 17

and squaring is well-defined in mod arithmetic: i.e.

if y = z mod 17 then y^2 = z^2 mod 17

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