Elementary Number Theory Problems 4.2 Solution (David M. Burton's 7th Edition) - Q17

My Solution for "Prove that whenever $ab \equiv cd \pmod {n}$ and $b \equiv d \pmod {n}$, with $gcd(b, n) = 1$, then $a \equiv c \pmod {n}$."


Table of Contents


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.


Prove that whenever $ab \equiv cd \pmod {n}$ and $b \equiv d \pmod {n}$, with $gcd(b, n) = 1$, then $a \equiv c \pmod {n}$.


As $ab \equiv cd \pmod {n}$ and $b \equiv d \pmod {n}$, we have $ab \equiv cb \pmod {n}$. Since $gcd(b, n) = 1$, from Corollary 1 of Theorem 4.3, $a \equiv c \pmod {n}$.

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