Elementary Number Theory Problems 4.2 Solution (David M. Burton's 7th Edition) - Q18

Background

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

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 $a \equiv b \pmod {n_{1}}$ and $a \equiv c \pmod {n_{2}}$, prove that $b \equiv c \pmod {n}$, where the integer $n = gcd(n_{1}, n_{2})$.

Solution