Elementary Number Theory Problems 4.2 Solution (David M. Burton's 7th Edition) - Q10

My Solution for "If $a_{1}, a_{2}, ..., a_{n}$ is a complete set of residues modulo $n$ and $gcd(a, n) = 1$, prove that $aa_{1}, aa_{2}, ... , aa_{n}$ is also a complete set of residues modulo $n$. [Hint: It suffices to show that the numbers in question are incongruent modulo $n$.]"

Ran
Ran


Background

All theorems, corollaries, and definitions listed in the book's order:

Theorems and Corollaries in Elementary Number Theory
All theorems and corollaries mentioned in David M. Burton’s Elementary Number Theory are listed by following the book’s order. (7th Edition) (Currently Ch 1 - 4)

I will only use theorems or facts that are proved before this question. So you will not see that I quote theorems or facts from the later chapters.

Question

If $a_{1}, a_{2}, ..., a_{n}$ is a complete set of residues modulo $n$ and $gcd(a, n) = 1$, prove that $aa_{1}, aa_{2}, ... , aa_{n}$ is also a complete set of residues modulo $n$.
[Hint: It suffices to show that the numbers in question are incongruent modulo $n$.]

Solution

We can prove this by contradiction by using the Observation From A Complete Set of Residues.

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