Theorem regarding mod properties

In summary, the conversation is discussing the theorem that states when two numbers are congruent (mod c), their sum or difference is also congruent (mod c). However, the conversation also brings up the concept of equivalence classes and how a number can have infinitely many elements in it in modulo arithmetic. There is also some confusion about using equal signs instead of congruent signs and the importance of starting from the definition of congruence.
  • #1
plusunim
3
0
There is a theorem regarding mod properties such that when a=a'(mod
c) and b=b'(mod c) then a*±b=a'±b' (mod c) right?

Now, 5=5 mod7. applying it to the above, 10=10 mod7, which is not
true since 3=10 mod7. Why? I'm confused.

Thanks
 
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  • #2
No 10 is congruent to 10 mod 7 and it is congruent to 3. You are mistaken only in thinking this is wrong. really you ought to be susing congruent signs, not equal signs. Any equivalence class has infinitely many elements in it in modulo arithmetic.
 
  • #3
howcome 10=10 mod7? I thought that you can have the same numbers on both sides only when the numbers are less than the mod. Am I wrong again?

-- and by the way, to type fast I use equal signs, I know it's not fully correct, please bear my momentaneous impatience --

Thanks for replying anyway
 
  • #4
Why would you only be able to have the same numbers on both sides if they were sufficiently small? x=y mod n means exactly that n divides x-y or equivalently that there is an integerk such that x=kn+y and that is all.
 
  • #5
plusunim said:
howcome 10=10 mod7? I thought that you can have the same numbers on both sides only when the numbers are less than the mod.
As matt has said already, start from the definition of congruence. You are confusing a residue with a least positive residue (commonly referred to as a remainder).

Again, start from the beginning and work your way up.
 
  • #6
yes I think so. I'll continue reading on. Thanks for the help :)
 

FAQ: Theorem regarding mod properties

1. What is the theorem regarding mod properties?

The theorem regarding mod properties states that for any two integers a and b, and a positive integer n, the following properties hold true:
a) (a + b) mod n = (a mod n + b mod n) mod n
b) (a * b) mod n = (a mod n * b mod n) mod n
c) (a - b) mod n = (a mod n - b mod n) mod n
d) (a/b) mod n = (a * b^-1) mod n, where b^-1 is the modular multiplicative inverse of b mod n.

2. What is the significance of the theorem regarding mod properties?

This theorem is important in number theory and modular arithmetic, as it allows for simplification and manipulation of expressions involving modular arithmetic. It also helps in solving equations involving modular congruences.

3. How is the theorem regarding mod properties used in cryptography?

In cryptography, the theorem is used to encrypt and decrypt messages using modular arithmetic operations. It is also used in the generation and verification of digital signatures.

4. Can the theorem be extended to non-integer values?

No, the theorem only holds for integers and cannot be extended to non-integer values.

5. Are there any exceptions to the theorem regarding mod properties?

Yes, the theorem does not hold true if the integers a and b are not relatively prime to n. In such cases, the properties may not hold and may result in incorrect calculations.

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