15 questions linked to/from Not understanding Simple Modulus Congruency
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### Iterations of modulus operation

While working on a completely unrelated task, I thought up the following problem: Consider the following process. Let $a_0$ and $n$ be given, and determine $a_1,\ldots, a_k$ as follows: a_{j+1}...
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### Euclid proof explanation

Can anyone help me understand the following proof that if $p|ab$ then $p|a$ or $p|b$? This proof is on a separate question. Suppose there were a counterexample, with $pa=bc$, $p$ a prime, but ...
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### Find $a \in \mathbb Z$, which solves $11 \cdot a \equiv 1 \pmod {1247}$

How would you solve the question (in the title)? Can I apply your approach/solution (for the title question) also for: $13 \cdot a \equiv 1 \pmod {1337}$ and $69 \cdot a \equiv 8 \pmod {8008}$ etc.?
Normal Euclidean algorithm iterates $(a$ mod $b, b)$ for $a>b$. I have noticed that iterating $(a$ mod $b, a)$ (keeping the $a$ fixed through descent) will also give the GCD. Can anyone show why ...