The discussion revolves around simplifying congruences in modular arithmetic, specifically focusing on how to reduce a polynomial expression modulo 25. Participants are examining the transformation of the expression 125t³ + 75t² + 50t + 4 into 65t + 5 under modulo 25.
There is an active exploration of the concepts involved in modular arithmetic, with participants seeking clarification on the implications of congruences. Some have provided insights into the relationship between the coefficients and the modulus, while others express confusion about the variables and their roles in the simplification process.
Participants are working within the constraints of a homework assignment, which may limit the information available for discussion. There is a focus on understanding the definitions and properties of modular arithmetic without providing direct solutions.
125\equiv 0\ (\!\!\!\!\mod25)Instinctlol said: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 said:125\equiv 0\ (\!\!\!\!\mod25)
etc.
Instinctlol said:I don't understand please clarify
micromass said:Do you understand what (mod 25) means?
Instinctlol said: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)Instinctlol said:I don't understand where he got the 125 and what happened to the t
SammyS said:If \ 125\equiv 0\ (\!\!\!\!\mod25)\, then \ 125t^3\equiv 0t^3\ (\!\!\!\!\mod25)
You do understand that \ 125\equiv 0\ (\!\!\!\!\mod25)\,,\ don't you?
Anyway ...Instinctlol said:That means 25|125, sorry I guess the variables just confuses me.
SammyS said: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 \ .
Well, the remainder of 65t+5 is the same as the remainder of 250t3+150t2+65t+5 when divided by 25 .Instinctlol said:Is it because 25|250 and 25|150 so the remainder is 65t+5?