# how to slove the congruence s.t x^2+1=0 mod 5 ??

• May 18th 2009, 01:40 AM
xixihaha
how to slove the congruence s.t x^2+1=0 mod 5 ??
how to slove the congruence such that

$\displaystyle x^{2}+1\equiv 0 (\bmod 5)$

cheers!
• May 18th 2009, 04:29 AM
Since you're working in mod 5 you just need to check the values 0,1,2,3,4.

For 0 we have, $\displaystyle 0^2 + 1 = 1$
For 1 we have, $\displaystyle 1^2 + 1 = 2$
For 2 we have, $\displaystyle 2^2 + 1 = 5 = 0$ mod 5
For 3 we have, $\displaystyle 3^2 + 1 = 10 = 0$ mod 5
For 4 we have, $\displaystyle 4^2 + 1 = 17 = 2$ mod 5

So x = 2 or 3.

Do you understand what mod 5 means?
• May 18th 2009, 05:05 AM
xixihaha
if the modulo here is 7. shall i only need to consider the values 0 up to 6?
• May 18th 2009, 05:07 AM
Rapha
Hi

Quote:

Originally Posted by xixihaha
if the modulo here is 7. shall i only need to consider the values 0 up to 6?

Yep, that is correct.

Yours
Rapha

Edit:

Quote:

see if the question is like x^2-1= 0 mod 5
(edited a "0" )

Yes you can use the same method.
• May 18th 2009, 05:33 AM
xixihaha
thank you~i understand now.

see if the question is like
$\displaystyle x^{2}-1\equiv 0 (\bmod 5)$
can i use that method to solve the problem??
• May 18th 2009, 05:39 AM
Yep. Just plug in the values of x you can use and mod the answer by 5 until its between (or equal to) 0 and 4. Can you figure out which values of x will give $\displaystyle x^2 - 1 = 0$ mod 5? Hint: there's two of them.
• May 18th 2009, 05:48 AM
xixihaha
$\displaystyle (x-1)(x+1)\equiv 0 (\bmod 5)$

$\displaystyle x\equiv 1 (\bmod 5)$

$\displaystyle x\equiv -1 (\bmod 5)$

so x =0,1

am i right?
• May 18th 2009, 06:33 AM
I think you've mistyped there. Should be 'so x = -1,1'

But -1 = 4 mod 5 so the answer is x = 1 and 4.

Test this by putting the values into $\displaystyle x^2 - 1$.

x=0 gives 0-1 = -1 = 4 mod 5 (since -1 + 5 = 5)
x=1 gives 1-1 = 0.
x=2 gives 4-1 = 3.
x=3 gives 9-1 = 8 = 3 mod 5 (since 8 - 3 = 5)
x=4 gives 16-1 = 15 = 0 mod 5 (since 15 - 5 - 5 - 5 = 0)

This confirms that 1 and 4 are the correct answers.
• May 18th 2009, 07:35 AM
xixihaha
oh yes. you are right. im quite clear now.
last,i'll say thank you to Rapha and Deadstar for helping me.
cheers.