Homework Help: Question about simplifying congruences

1. Nov 14, 2012

Instinctlol

1. The problem statement, all variables and given/known data
I am looking at an example from my book and I have no clue how this is done.

2. Relevant equations
I solved the left hand side and got this
125t3+75t2+50t+4

How did they reduce it to 65t+5?

3. The attempt at a solution
http://i48.tinypic.com/2v2gpap.jpg

2. Nov 15, 2012

SammyS

Staff Emeritus
$125\equiv 0\ (\!\!\!\!\mod25)$

etc.

3. Nov 15, 2012

Instinctlol

4. Nov 15, 2012

micromass

Do you understand what (mod 25) means?

5. Nov 15, 2012

Instinctlol

I don't understand where he got the 125 and what happened to the t

6. Nov 15, 2012

micromass

$$125t^3+75t^2+50t+4 = 65t+5 ~\text{(mod 25)}$$

right??

So, I ask: do you know what (mod 25) means?

7. Nov 15, 2012

SammyS

Staff Emeritus
If $\ 125\equiv 0\ (\!\!\!\!\mod25)\$, then $\ 125t^3\equiv 0t^3\ (\!\!\!\!\mod25)$

You do understand that $\ 125\equiv 0\ (\!\!\!\!\mod25)\,,\$ don't you?

8. Nov 15, 2012

Instinctlol

That means 25|125, sorry I guess the variables just confuses me.

9. Nov 15, 2012

SammyS

Staff Emeritus
Anyway ...

Do you now understand why $\displaystyle \ \ 250 t^3+150 t^2+65 t+ 5 \equiv 65t+5 ~\text{(mod 25)}\ ?$

Yes, I get $\displaystyle \ \ 2(1+5t)^3+7(1+5t)-4= 250 t^3+150 t^2+65 t+ 5 \ .$ ​

10. Nov 15, 2012

Instinctlol

Is it because 25|250 and 25|150 so the remainder is 65t+5?

11. Nov 15, 2012

SammyS

Staff Emeritus
Well, the remainder of 65t+5 is the same as the remainder of 250t3+150t2+65t+5 when divided by 25 .