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4

I will answer with an example. I seek the Bezout coefficients for 99 and 707. First I execute the Euclidean algorithm: \begin{align*} 707 &= 7 \cdot 99+14\\ 99 &= 7 \cdot 14+ 1 \end{align*} Now, I will recursively "backtrack"  \begin{align*} 1 &= 99-7\cdot 14\\ &= 99-7 \cdot (707-7 \cdot 99)\\ &=50 \cdot 99 - 7 \...

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Simplest explanation I've seen is to take $a, b$ integers and consider the set $\{u a + v b\}$ for integer $u, v$. It is a bunch of integers, so it has to contain a smallest positive one, call it $d = u_0 a + v_0 b$. Now divide $a$ by $d$: $a = q d + r$, by the Euclidean "algorithm" $0 \le r < d$. You see that: \$\begin{align*} a &= (...

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Benjamin Hutz has a recent book that could be appropriate: An Experimental Introduction to Number Theory. This book presents material suitable for an undergraduate course in elementary number theory from a computational perspective. It seeks to not only introduce students to the standard topics in elementary number theory, such as prime factorization and ...

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