MathDB

Let

n

be a positive integer,

n\geq 2

. Let

M=\{0,1,2,\ldots n-1\}

. For an integer nonzero number

a

we define the function

f_{a}: M\longrightarrow M

, such that

f_{a}(x)

is the remainder when dividing

ax

n

. Find a necessary and sufficient condition such that

f_{a}

is bijective. And if

f_{a}

is bijective and

n

is a prime number, prove that

a^{n(n-1)}-1

is divisible by

n^{2}

Problem Statement