What solutions does the equation $4x≡2\mod 10$ have? (Hint, it will have more that one.)  What about solutions to an equation $ax≡d \mod n$, where $d=\gcd(a,n)$? This is the question that appear in our universities last year paper. But it is seems very difficult for me like for first part I am able to solve by simplifications I got $x=5.Z+3$ as solution which makes sense and seems fair but for the second one I have no idea how can I approach that. Can you please help me out with that one. I tried to simplify that by using Euclidean algorithm but still I didn't get any idea that what actually the solution will be. My final exam is on 2nd of next month and because of covid I am unable to contact in person for any help. So any help from this site will be appreciated. Thanks you for looking at this question!
What solutions does the equation $4x\equiv2\mod10$ have? (Hint, it will have more that one.) What about solutions to an equation $ax\equiv d\mod n$, where $d=\gcd(a,n)$?  This is the question that appeared in our universities last year paper. But it seems very difficult for me like for first part I am able to solve by simplifications I got $x=5\cdot Z +3$ as solution which makes sense and seems fair but for the second one, I have no idea how can I approach that. Can you please help me out with that one. I tried to simplify that by using Euclidean algorithm but still, I didn't get any idea that what actually the solution will be. My final exam is on 2nd of next month and because of covid I am unable to contact in person for any help. So any help from this site will be appreciated. Thank you for looking at this question!
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