@Roddy macphee hi.

hello

I alway find it ironic, the most advanced people, never have the easy links back to grade school math.

mod doesn't care about the multiplier though.

OK.

Yes.

One definition of mod m, is that y is congruent to b mod m only if the difference y-b is divisible by m

addition table mod 10 coming soon.

OK I do not know that. @RoddyMacPhee

$$\begin{array}{c|c|c|c|c|c|c|c|c|c|}+&1&2&3&4&5&6&7&8&9&0\\\hline 1&2&3&4&5&6&7&8&9&0&1\\\hline 2&3&4&5&6&7&8&9&0&1&2\\\hline 3&4&5&6&7&8&9&0&1&2&3\\\hline 4&5&6&7&8&9&0&1&2&3&4\\\hline 5&6&7&8&9&0&1&2&3&4&5\\\hline 6&7&8&9&0&1&2&3&4&5&6\\\hline 7&8&9&0&1&2&3&4&5&6&7\\\hline 8&9&0&1&2&3&4&5&6&7&8\\\hline 9&0&1&2&3&4&5&6&7&8&9\\\hline 0&1&2&3&4&5&6&7&8&9&0\\\hline \end{array}$$

most of the rules of normal arithmetic work in mod.

$(1+3)+6= 4+6=0 \bmod 10$ for example.

questions ?

Wow.

I need some time to read all your post.

The multiplication table comes from this, by looking at repeated addition of the same amount: so 9+9=8 mod 10; 8+9=7 mod 10; etc.

sorry if my notation is bad. concepts are better aquainted first.

Yes, and I feel sorry how the other user behave with you.

@RoddyMacPhee

By the way hi.

yeah it's because one strictly limits the values it can take on. while the other just cares if they are in the same remainder class.

en.m.wikipedia.org/wiki/Modular_arithmetic

remander classes are the y-intercepts lesd than m

$$\begin{array}{c|c|c|c|c|c|c|c|c|c|}\times&1&2&3&4&5&6&7&8&9&0\\\hline 1&1&2&3&4&5&6&7&8&9&0\\\hline 2&2&4&6&8&0&2&4&6&8&0\\\hline 3&3&6&9&2&5&8&1&4&7&0\\\hline 4&4&8&2&6&0&4&8&2&6&0\\\hline 5&5&0&5&0&5&0&5&0&5&0\\\hline 6&6&2&8&4&0&6&2&8&4&0\\\hline 7&7&4&1&8&5&2&9&6&3&0\\\hline 8&8&6&4&2&0&8&6&4&2&0\\\hline 9&9&8&7&6&5&4&3&2&1&0\\\hline 0&0&0&0&0&0&0&0&0&0&0\\\hline \end{array}$$ is the multiplication table mod 10 (they change from modulus to modulus)

If you follow the pattern of repeated multiplication you'll find a surprise.

sorry if I'm going too fast.

exponentiation has the following twist: $$\begin{array}{c|c|c|c|c|c|c|c|c|c|c|}\text{^}&1&2&3&4&5&6&7&8&9&0\\\hline 1&1&2&3&4&5&6&7&8&9&0\\\hline 2&1&4&9&6&5&6&9&4&1&0\\\hline 3&1&8&7&4&5&6&3&2&9&0\\\hline 4&1&6&1&6&5&6&1&6&1&0\\\hline \\\hline 0&1&1&1&1&1&1&1&1&1&1\\\hline \end{array}$$

every column repeats if you extend this further.