# (Elementary number theory) Understanding congruence and modulus

• I
• Leo Liu
In summary: This makes sense, but could you explain why this expression still holds when a1 and b1 is followed by (mod m)?
Leo Liu
TL;DR Summary
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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:

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?

Leo Liu
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.

Leo Liu
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)##.

Leo Liu
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)?

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.

Leo Liu

## 1. What is congruence in elementary number theory?

Congruence in elementary number theory is a relation between two integers where they have the same remainders when divided by a third integer, known as the modulus. In other words, two integers are congruent if they have the same remainder when divided by the same number.

## 2. What is the purpose of studying congruence in elementary number theory?

Studying congruence in elementary number theory helps us understand the patterns and properties of integers. It also has many practical applications, such as in cryptography and computer science.

## 3. How is congruence represented in mathematical notation?

Congruence is represented using the symbol ≡ (three horizontal lines) and is read as "is congruent to". For example, a ≡ b (mod n) means that a is congruent to b modulo n.

## 4. What is the difference between congruence and equality?

Congruence and equality are both mathematical relationships, but they have different meanings. Congruence means that two numbers have the same remainders when divided by a third number, while equality means that two numbers are exactly the same.

## 5. How is the modulus used in congruence in elementary number theory?

The modulus is a crucial part of congruence in elementary number theory as it determines which numbers are congruent to each other. It is used to define the congruence relation and to perform calculations involving congruence.

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