hi there I having some troble with a few problems i was wondering if i could get some help?
x^2 equvilent to -1 mod 19
x^2 equvilent to -1 mod 19
4x equvilent to 6 mod 15
thanks for your help
I'll defer the first too to more able number theorists.
$\displaystyle 4x\equiv 6\text{ mod }15\implies 4x=6-15y\implies 4x+15y=6$ this is solvable since $\displaystyle 6=6(15,4)$. Now do you know how to solve Diophantine equations?
P.S. It might be easier to just guess :S
$\displaystyle (4,15)=1 \implies 4^{-1} $ exists modulo $\displaystyle 15 $.
You can find $\displaystyle 4^{-1} $ by the Euclidean algorithm.
I omit details but $\displaystyle 4\cdot4\equiv1\bmod{15}\implies 4^{-1}\equiv4\bmod{15} $.
So $\displaystyle x\equiv 4^{-1}\cdot4x\equiv 4^{-1}\cdot6\equiv 4\cdot6 = 24\equiv 9 \bmod{15} $