MATHmaniac aka CrUNchy

THEOREM 1
Let P be a prime number and suppose that P is not divisible by a.Then,

THEOREM 2
If P is a prime number,then for any integer a.

Now,Using Fermat's Theorem we're going to proof:

a)
Theorem 1 said, so,



.
.
.

thus,

simplify it,

How (P-1) become congruence to -1??
Ok,let's say P as 19
so,when (19-1) become 18,then


b)
THEOREM 2 said,
so,



.
.
.

thus,

simplify it,

How come
Again P as 19,thus

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Definition:
An arithmetic function is a function that defined for all positive integers

Definition:
An arithmetic function f is called multiplicative if f(mn) = f(m) f(n) whenever m and n are relatively prime positive integers.It is called multilplicative if
f(mn) = f(m) f(n) for all positive integers m and n.

Let's see...
and are multiplicative function.
Let say . Find and .


Positive divisor of n=

by multiplying P,



thus,


Get it??
Let's use number,..






thus,
Let say k=5,so

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DEFINITION
Given a positive integer n, let denote the number of positive divisor of n and denote the sum of this divisors.

• - The number of positive divisor of n
• - The sum of the positive divisor of n

Example:

1. n=6
   The divisions of 6 are 1,2,3,6
   So, T (n) = 4, σ (n) = 12
2. n=12
   The divisions of 12 are 1,2,3,4,6,12
   So, T (n) = 6, σ (n) = 28
3. n=100
   the divisions of 100 are 1,2,4,5,10,20,25,50,100
   So, T (n) = 9, σ (n) = 217
   
   

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