[itex]125\equiv 0\ (\!\!\!\!\mod25) [/itex]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:[itex]125\equiv 0\ (\!\!\!\!\mod25) [/itex]
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 [itex]\ 125\equiv 0\ (\!\!\!\!\mod25)\ [/itex], then [itex]\ 125t^3\equiv 0t^3\ (\!\!\!\!\mod25)[/itex]Instinctlol said:I don't understand where he got the 125 and what happened to the t
SammyS said:If [itex]\ 125\equiv 0\ (\!\!\!\!\mod25)\ [/itex], then [itex]\ 125t^3\equiv 0t^3\ (\!\!\!\!\mod25)[/itex]
You do understand that [itex]\ 125\equiv 0\ (\!\!\!\!\mod25)\,,\ [/itex] 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 [itex]\displaystyle \ \ 250 t^3+150 t^2+65 t+ 5 \equiv 65t+5 ~\text{(mod 25)}\ ?[/itex]
Yes, I get [itex]\displaystyle \ \ 2(1+5t)^3+7(1+5t)-4= 250 t^3+150 t^2+65 t+ 5 \ .[/itex]
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?
A congruence is a mathematical relationship between two numbers or objects, where they have the same remainder when divided by a certain number. It can also be thought of as two numbers being "equivalent" in a specific context.
Simplifying congruences allows us to solve for unknown values and make mathematical operations easier. It also helps us to identify patterns and relationships between numbers.
To simplify a congruence, we use the modulus operator (represented by the symbol %) to find the remainder of the given numbers when divided by the modulus (the number after the % symbol). This remainder is then used to rewrite the congruence in a simpler form.
For example, if we have the congruence 12 ≡ 2 (mod 5), we can simplify it by finding the remainder of 12 when divided by 5, which is 2. This gives us the simplified congruence 2 ≡ 2 (mod 5).
Simplifying congruences is used in various fields, such as number theory, cryptography, and computer science. It is also used in solving problems involving modular arithmetic, such as finding the day of the week for a given date or calculating the last digit of a large number.