Counter examples as Mnemonics - Remainders

When refreshing my memory about a Mathematical subject - I find counter examples can be a useful way of getting active. Here is an example about mod() / Congruences.

Working mod 4 with two numbers a=5 and b=7

...and writing in computing function form we say...

mod(a,4) = 1

  and

mod(b,4) = 3

Now apply a scaling (k) to both numbers ... say ... scaling by 11

mod(ka,4) = 3                       (ka is 55)

  and

mod(kb,4) = 1                       (kb is 77)


And what we observe is that the results seem to have exchanged places :)

This counter example is a good reminder that applying the same scaling to two different numbers does not always preserve the mod result.

There is much theory to access regarding congruences / modulo, and counter examples like this, can be a good way of stimulating a refresher of that theory.