1. Remainders again

find the remainder when $\displaystyle 1992^{1998} - 1955^{1998} - 1938^{1998} + 1901^{1998}$ is divided by $\displaystyle 1998$

i tried using eulers theorem and got only this much

$\displaystyle 1992^{1980}\equiv 1mod 1998$
similarly for oders...but cudnt get after that

so i need a kind help.

thanku

2. first of all can anyone tell me if this q uses any modular arithmetic..

bec its getting very untidy with congruency...

3. Originally Posted by banku12
find the remainder when $\displaystyle 1992^{1998} - 1955^{1998} - 1938^{1998} + 1901^{1998}$ is divided by $\displaystyle 1998$

i tried using eulers theorem and got only this much

$\displaystyle 1992^{1980}\equiv 1mod 1998$
similarly for oders...but cudnt get after that

so i need a kind help.

thanku
If you got that they are all congruent to one then wouldnt the expression be congruent to -2=1996?

4. Originally Posted by banku12
find the remainder when $\displaystyle 1992^{1998} - 1955^{1998} - 1938^{1998} + 1901^{1998}$ is divided by $\displaystyle 1998$

i tried using eulers theorem and got only this much

$\displaystyle 1992^{1980}\equiv 1mod 1998$
similarly for oders...but cudnt get after that

so i need a kind help.

thanku
$\displaystyle 1992^{1980}\equiv 1296\mod{1998}$

Here's a hint: $\displaystyle \phi(1998) = 648$ and $\displaystyle 1998 = 3\cdot648+54$.

5. seriously not getting

can any one of u plz post a complete soln

6. I should have mentioned in my last hint that for Euler's Theorem to work i.e. $\displaystyle a^{\phi(n)}\equiv1\mod{n}$, we need $\displaystyle (a,n)=1$. So Euler's Theorem doesn't apply to every term here.

Try this:

$\displaystyle 1992\equiv-6\mod{1998}$

So $\displaystyle 1992^{1998}\equiv6^{1998}\mod{1998}$.

Now do something similar to this: http://www.mathhelpforum.com/math-he...remainder.html