Square root of 1 with mod how to prove it?

In summary, the square root of 1 with mod is any number that, when squared and taken the modulus of 1, equals 1. This can be proven by using the definition of the modulus function, which only returns positive values and therefore only allows 1 as a square root of 1 with mod. This concept is related to congruence, as it represents all numbers that are congruent to 1 when taken the modulus of 1. Finally, the square root of 1 with mod is a unique value, as there can only be one number that satisfies the conditions.
  • #1
jacquelinek
3
0
I know that if n is odd and has k distinct prime factors, then the number of roots, x^2 = 1 (mod n), is equal to 2^k.
However, I don't know how to give a formal proof to it.
I simply want to bypass the generalized form x^2 = a (mod n).
How can I prove it directly?
Thank you.
 
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  • #2
Hmm... assuming it holds mod p^k, you could just CRT the results together.
 

FAQ: Square root of 1 with mod how to prove it?

What is the square root of 1 with mod?

The square root of 1 with mod is any number that, when squared and taken the modulus of 1, equals 1.

How can you prove that the square root of 1 with mod is 1?

To prove that the square root of 1 with mod is 1, we can use the definition of the modulus function, which states that the modulus of a number is the remainder after division by that number. Since 1 divided by any number will always have a remainder of 1, the square root of 1 with mod will always be 1.

Can there be any other square root of 1 with mod besides 1?

No, there cannot be any other square root of 1 with mod besides 1. This is because the modulus function only returns positive values, and the only positive number that, when squared, results in 1 is 1 itself.

How does the square root of 1 with mod relate to the concept of congruence?

The square root of 1 with mod relates to congruence in the sense that it represents all numbers that are congruent to 1 when taken the modulus of 1. In other words, any number that gives a remainder of 1 when divided by 1 can be considered a square root of 1 with mod.

Is the square root of 1 with mod a unique value?

Yes, the square root of 1 with mod is a unique value. As mentioned before, the only number that is congruent to 1 when taken the modulus of 1 is 1 itself. Therefore, there can only be one square root of 1 with mod, making it a unique value.

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