Let m, n, r, e be pos. integers. I am trying to show that given m = n (mod e*r) that m^e=n^e (mod r*e^2). I have tried using the binomial theorem but keep running into problems.

Could someone help please?? thanks!

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- May 7th 2011, 05:53 PMarlingtonbassettcongruence question
Let m, n, r, e be pos. integers. I am trying to show that given m = n (mod e*r) that m^e=n^e (mod r*e^2). I have tried using the binomial theorem but keep running into problems.

Could someone help please?? thanks! - May 7th 2011, 07:39 PMtopsquark
This is a little rough, but perhaps you can clean it up and make everything neat.

The first condition says

$\displaystyle m \equiv n \text{ (mod (er))}$

So we know that

m = a(er) + b

n = c(er) + d

where a, b, c, d are integers and 0 <= b < er and 0 <= d < er. Since m = n (mod er) we know that b = d.

Now, let's form m^e.

$\displaystyle m^e = (a(er) + b)^e = a^e (er)^e + e a^{e - 1}(er)^{e - 1}b + \text{ ... } + e a(er)b^{e-1} + b^e$

Note that every term but the last is proportional to e^2r. Thus m^e has the form:

$\displaystyle m^e = p(e^2r) + b^e$

where p is an integer. We don't need the usual 0 <= b^e < e^2 r condition as you will see.

Now do the same for n^e. Then use the fact that b = d from the initial condition.

-Dan - May 8th 2011, 08:15 AMarlingtonbassett
thanks for that. it all makes sense except two of the beginning lines...how did you get

m=aer+b and n= cer+d?

everything else makes sense to me.

thanks! - May 8th 2011, 09:59 AMtopsquark
- May 8th 2011, 10:09 AMarlingtonbassett
nm got it thanks again for the help