# Solving System of Single-Variable Modular Equations

I have been scratching my head over the following problem for a couple days now. How do you solve for α in the system of expressions below?

$$\frac 89 α≡40\pmod{360}$$ $$\frac 8{11} α≡0\pmod{360}$$ $$α≡0 \pmod{360}$$

I know the solution should be 15840, but how do I go about solving this? From the last 2 expressions I can conclude that the answer should be a multiple of 3960 (expression 2 implies α is some multiple of 495, and expression 3 implies α is a multiple of 360, and the LCM of these 2 numbers is 3960). How do I reach the final answer of 15840 other than by just trying multiples of 3960 until I get one to work?

(edit) I mean to say the solution is integer multiples of 15840, since any multiple of this number will satisfy these 3 constraints.

• Why does the second equation imply $\alpha$ is a multiple of $495$? Substituting $\alpha \equiv 0 \pmod{360}$ works fine, no? – TMM Sep 29 '16 at 18:05
• When you multiply both sides of the congruence by 11/8, then it becomes α≡0(mod495), indicating α is some integer multiple of 495. – Jared Sep 29 '16 at 18:31

I finally figured it out on my own. The key for me was finding out about the Euclidean Algorithm and how it can be used to solve modular arithmetic problems. First I rewrote the last 2 expressions, where I found the answer would be some multiple of 3960. $$α≡0\pmod{360},α≡0\pmod{495}\Rightarrowα≡0\pmod{3960}$$ $$3960|α\Rightarrowα=3960x$$ Then I rewrote the first expression by multiplying both sides by 9/8. $$\frac89α≡40\pmod{360}\Rightarrowα≡45\pmod{405}$$ $$405|(α-45)\Rightarrowα=45+405t$$ Lastly, I set these two quantizations of α equal to one another to get one more expression. $$3960x=45+405t$$ I used this final expression and the Euclidean Multiplicative Inverse Algorithm to set up some substitutions that could be used to solve for x and t. Turned out to be a little more involved than I anticipated, but it was a pretty nifty trick to use to solve these.