I (Elementary number theory) Understanding congruence and modulus

AI Thread Summary
The discussion focuses on understanding congruence and modulus in elementary number theory, specifically how to apply the transitivity property and the congruence addition and multiplication rules. Participants clarify that if two numbers are congruent modulo a certain number, one can replace one number with its equivalent in calculations involving sums or products. The example of 9 being congruent to 2 modulo 7 is used to illustrate this principle. Additionally, the congruence rules allow for the manipulation of expressions involving modular arithmetic, ensuring that the equivalence holds true in calculations. Overall, the conversation emphasizes the importance of understanding these properties for solving congruence problems effectively.
Leo Liu
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In this question, how does the step marked with 1 become the step marked with 2? I can see that the transitivity property of congruence is used, but I don’t know what exactly is going on here. Can someone please explain? Also at which step is Congruence Add and Multiply used?

Thanks.

Preposition used:
Screen Shot 2021-11-15 at 9.01.09 PM.png
 
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Isn’t 9 mod 7 the value 2?

Similarly for 3^3 is 27 which is one less than 28.

what rule would they be using to do this?
 
jedishrfu said:
Isn’t 9 mod 7 the value 2?

Similarly for 3^3 is 27 which is one less than 28.
Yes. But I just don't understand how the theorem is applied here.
 
The theorem says the modulo N of a sum or product of a list of numbers is the same as the sum or product of the modulos of those numbers, right?
 
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jedishrfu said:
The theorem says the modulo N of a sum or product of a list of numbers is the same as the sum or product of the modulos of those numbers, right?
Right. But I am afraid that I still cannot see why as the congruence equations are not in the same form as the preposition.
 
Leo Liu said:
Right. But I am afraid that I still cannot see why as the congruence equations are not in the same form as the preposition.
What does that mean?

We have ##9 = 2 + 7##. Therefore ##9 \equiv 2 \ (mod \ 7)##. Therefore we may replace ##9## by ##2## in any equation modulo ##7##. That's the idea.
 
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The proposition says that if ##a_1\equiv a_2 \; (mod\; m)## and ##b_1\equiv b_2 \; (mod\; m)##, then you have ##a_1b_1\equiv a_2b_2 \; (mod\; m)##.

In the example it is applied to ##9 \equiv 2 \; (mod\; 7)## and ##3^3 \equiv -1 \; (mod\; 7)##.
 
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PeroK said:
What does that mean?

We have ##9 = 2 + 7##. Therefore ##9 \equiv 2 \ (mod \ 7)##. Therefore we may replace ##9## by ##2## in any equation modulo ##7##. That's the idea.
Thanks. You answer helped me understand it intuitively. Yet I still don't know how the congruence add and multiply is applied to this step.
 
martinbn said:
The proposition says that if ##a_1\equiv a_2 \; (mod\; m)## and ##b_1\equiv b_2 \; (mod\; m)##, then you have ##a_1b_1\equiv a_2b_2 \; (mod\; m)##.

In the example it is applied to ##9 \equiv 2 \; (mod\; 7)## and ##3^3 \equiv -1 \; (mod\; 7)##.
This makes sense, but could you explain why this expression still holds when a1 and b1 is followed by (mod m)?
 
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Leo Liu said:
Thanks. You answer helped me understand it intuitively. Yet I still don't know how the congruence add and multiply is applied to this step.
Proposition 3 that you quoted in your OP is a formal way to say you can replace a number by any other number of modular equivalence in products and sums. That's what it means - and Proposition 3 is that idea written out formally.
 
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