Question about simplifying congruences

[Instinctlol]

Homework Statement

I am looking at an example from my book and I have no clue how this is done.

Homework Equations

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

How did they reduce it to 65t+5?

The Attempt at a Solution

http://i48.tinypic.com/2v2gpap.jpg

 

[SammyS]

Homework Statement

I am looking at an example from my book and I have no clue how this is done.

Homework Equations

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

How did they reduce it to 65t+5?

The Attempt at a Solution

http://i48.tinypic.com/2v2gpap.jpg

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

etc.

 

[Instinctlol]

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

etc.

I don't understand please clarify

 

[micromass]

I don't understand please clarify

Do you understand what (mod 25) means?

 

[Instinctlol]

Do you understand what (mod 25) means?

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

 

[micromass]

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

But you're asking why

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

right??

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

 

[SammyS]

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

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

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

 

[Instinctlol]

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

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

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

 

[SammyS]

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

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

 

[Instinctlol]

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

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

 

[SammyS]

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

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