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This is my first time posting here so apologies if I miss out any etiquette!

I'm working my way through certain sections of Basic Mathematics by Serge Lang as a preparation for a Stats course that I'm taking later in the year. My working level at the moment is pretty primitive, having been away from the world of maths for many years. I'm looking at the section on divisibility of integers and have come across a problem that I can't grasp the proof to.

The problem is as follows:

Let $a,b$ be integers.

Prove that if $a \equiv b \pmod{5} $ and $x \equiv y \pmod{5}$, then $ax \equiv by \pmod{5}$

The proof to the first section is clear enough, but I just can't understand the proof to the second section - there must be some implicit assumptions that I'm not following. The proof is as follows:

There exist integers $k,q$ such that $a = b+5k$ and $x = y+5q$. Then \begin{align} ax &= (b+5k)(y+5q) \\ &= by+5(ky+bq+5kq) \end{align}.

Then setting $t = ky+bq+5kq$ gives $ax=by+5t$, i.e. $ax \equiv by \pmod{5}$

I can't see how that leads to a proof. If anyone could unpack this for me I would be most grateful!

Many thanks.

Later
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Sam H
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    Please use MathJax to format your equations. – Gary Feb 03 '22 at 00:35
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    Which step specifically do you stop following? Expanding $(b+5k)(y+5q)$ is just a matter of algebra. Rearranging and factoring out a 5 from those terms with a factor of five is as well. Choosing to do so was to show that the distance between $ax$ and $by$ is a multiple of five which is precisely what we were asked to show in different words. – JMoravitz Feb 03 '22 at 01:10
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    By definition $,ax\equiv by\pmod{5}$ is true iff $,ax = by+ 5n,$ for some integer $n$. The proof shows this is true for $,n = ky+bq+5kq,,$ which is clearly an integer since $,k,y,b,q,5,$ are all integers, and integers are closed under sums and products. See this answer for basic congruence arithmetic laws, including this Congruence Product Rule, and see the Linked questions there for many worked examples. – Bill Dubuque Feb 03 '22 at 11:07

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