Distribution of power congruence classes

[dec178]

Hi, I need help to prove this for my professor
this is called "Distribution of power congruence classes" or something like that

With all n∈NiS∈N correct
1) n ≡Qs(n)(mod 10s-1)
2) n ≡Qs(n)(mod 10s+1)

http://img546.imageshack.us/img546/8341/withall.png

 

[Stephen Tashi]

Your question isn't clear.

You must explain your notation. What is $N_i$? What is $Q_s(n)$? What is $Q'_s(n)$ ?

Instead of "correct", perhaps you mean "it is true that".

 

[dec178]

Yes, I need to proove, that this is correct.
To separate $Q_s(n)$ and $Q'_s(n)$, I used apostrophe '
I don't know, professor just gave this for us in a middle of Modular arithmetic class

 

[Norwegian]

Can we perhaps decipher the question as follows:

Let n and s be positive integers, let Q_s(n) be the sum of the numbers formed by the digits of n in groups of s, starting from the right, and let Q_s'(n) be the alternating such sum.

Show that Q_s(n)$\equiv$n (mod 10^s-1) and Q_s'(n)$\equiv$n (mod 10^s+1)

 

[blue_raver22]

Dear student,

Thank you for reaching out for help with your professor's request. I am happy to assist you in proving the concept of "Distribution of power congruence classes."

Firstly, let us define what "power congruence classes" mean. In number theory, congruence is a relation between two integers where they have the same remainder when divided by a given number. For example, 15 ≡ 3 (mod 6) means that both 15 and 3 have a remainder of 3 when divided by 6.

Now, let us consider the power of a number. The power of a number is the number of times it is multiplied by itself. For instance, 3^2 = 9, where 3 is multiplied by itself 2 times.

Combining these two concepts, we can define power congruence classes as a set of numbers that have the same remainder when raised to a certain power and divided by a given number.

In your statement, you have provided two equations:
1) n ≡Qs(n)(mod 10s-1)
2) n ≡Qs(n)(mod 10s+1)

From these equations, we can deduce that the power (Qs(n)) of n is being taken modulo (mod) 10s-1 and 10s+1. This means that the remainder of n when raised to the power Qs(n) and divided by 10s-1 and 10s+1 will be the same.

To prove this, let us take an example.
Let n = 3, Qs(n) = 2, and s = 2.
Then, 3^2 ≡ 3 (mod 10^2-1) and 3^2 ≡ 3 (mod 10^2+1).
This means that both equations hold true, and the remainder of 3^2 when divided by 10^2-1 and 10^2+1 is 3.

We can also generalize this for any n, Qs(n), and s. This is because the congruence relation is transitive, i.e., if a ≡ b (mod m) and b ≡ c (mod m), then a ≡ c (mod m).

In conclusion, the "Distribution of power congruence classes" is a valid concept, and it holds