How do congruences work in mod arithmetic?

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In summary, the conversation discusses examples from a book using the property of congruences, specifically the property that states if a number is congruent to another number modulo m, then adding or subtracting a constant from both numbers will result in the same congruence. The examples illustrate this property by showing how 19 can be equivalent to 3 and how adding or subtracting a constant can maintain the congruence.
  • #1
mickles
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Hi, this is not a homework problem, i just have a hard time following the sequence of this

In the book , it shows a couple examples

(=== is the triple equal sign)

1) 26=19+7===3+7===10(mod8)
2) 15=19-4===3-4=-1(mod8)
3)38 = 19*2===3*2=6(mod8)
4) 7===2(mod5), 343=7^3===2^3=8(mod5)

I understand mod8 and whatnot, just how does the book go from 19+7 to 3+7, and from 19-4 to 3-4. I just don't get how 19 and 3 are logically connected

I see how 15===-1(mod8) and 26===10(mod8) and 38=6(mod8).

Any help understanding is appreciated
 
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  • #2
The examples you cited are using the following property of congruences: If [itex]a \equiv b \pmod{m}[/itex], then [itex] a + c \equiv b + c \pmod{m}[/itex]. So, in your first example, let a = 19, b = 3, c = 7 and m = 8.

Does that help?
 
  • #3
mickles said:
Hi, this is not a homework problem, i just have a hard time following the sequence of this

In the book , it shows a couple examples

(=== is the triple equal sign)

1) 26=19+7===3+7===10(mod8)
2) 15=19-4===3-4=-1(mod8)
3)38 = 19*2===3*2=6(mod8)
4) 7===2(mod5), 343=7^3===2^3=8(mod5)

I understand mod8 and whatnot, just how does the book go from 19+7 to 3+7, and from 19-4 to 3-4. I just don't get how 19 and 3 are logically connected

I see how 15===-1(mod8) and 26===10(mod8) and 38=6(mod8).

Any help understanding is appreciated
Any number A === A - N Mod N Thus 19 === 11 === 3 Mod 8, Therefore 19+7 === 3+7 and 19-4 === 3-4.
 
  • #4
Petek said:
The examples you cited are using the following property of congruences: If [itex]a \equiv b \pmod{m}[/itex], then [itex] a + c \equiv b + c \pmod{m}[/itex]. So, in your first example, let a = 19, b = 3, c = 7 and m = 8.

Does that help?

Yes that makes a lot more sense now with a,b,c, and m after looking at the theorem.

Thanks for you help
 
  • #5
ramsey2879 said:
Any number A === A - N Mod N Thus 19 === 11 === 3 Mod 8, Therefore 19+7 === 3+7 and 19-4 === 3-4.

thank you this also helped
 

1. What is Basic Congruences Confusion?

Basic Congruences Confusion is a concept in mathematics that deals with the equivalence of two numbers in terms of a specified modulus. It is commonly used in number theory and cryptography.

2. How is Basic Congruences Confusion used in cryptography?

Basic Congruences Confusion is used in cryptography to ensure the security of encrypted messages. By using a specified modulus, it becomes difficult for an attacker to determine the original message from the encrypted one.

3. How do I solve Basic Congruences Confusion problems?

To solve Basic Congruences Confusion problems, you need to use modular arithmetic. This involves finding the remainder when dividing the numbers by the specified modulus and then using this remainder to determine the equivalence of the two numbers.

4. Can Basic Congruences Confusion be applied to any numbers?

Yes, Basic Congruences Confusion can be applied to any numbers as long as a modulus is specified. However, it is most commonly used with whole numbers and integers.

5. What are some real-world applications of Basic Congruences Confusion?

Basic Congruences Confusion has many real-world applications, such as in secure communication systems, credit card encryption, and digital signatures. It is also used in the creation of pseudorandom number generators and in error-correcting codes.

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