HOW TO CHOOSE MOD

[MATHPRITOM]

"Prove the equation has no solution,has infinity solutions,has exactly 1 solution",we often get this type of problems in number theory.we often solve them,by using,mod.but,it is too hard to choose the correct number to do the modular equation.how to choose the correct mod???????

[Moon]

Choosing mod is more like art, not science! With some experience you can understand which mod to take. Important ones are $2,3,4,8$ for quadratic cases.

[Masum]

"Prove the equation has no solution,has infinity solutions,has exactly 1 solution",we often get this type of problems in number theory.we often solve them,by using,mod.but,it is too hard to choose the correct number to do the modular equation.how to choose the correct mod???????

Hmm. I should give an example to convince you. Certainly you have seen the equation \[x^2+y^2=123\] has no solution taking $\pmod 4$. Now, consider the equation \[x^2+y^2=794\]
The possibility is not ruled out taking $\pmod4$. But you don't get the solutions either. But note that $\pmod5$ comes to the rescue. Since $a^2\equiv0,\pm1\pmod5$, we have one of $x,y$ is divisible by $5$, say $x=5k$. Also since $x^2<794$, we have $x\le27$. Then possible values for $x$ are $5,10,15,20,25$. An easy check shows that $x=25$ satisfies. Hope you have a better idea now, and the statement of Moon is clarified a lot.

[Masum]

And remember, it is not necessary that you always need to choose a numeric mod, you may choose a mod with the variables in the equations or primes as well.

See this topic for some practice. I may add some more problems.

viewtopic.php?f=26&t=1570

[MATHPRITOM]

Thanks Masum vaiya ... a lot.