A proof about congruences got me wondering about the linearity property:
for
Here's how it went:
If (mod m) and (mod m), then (mod m)
By assumption and .
By the linearity property, . Re-arranging we get
Can you just rearrange that easily??? Sure the total sum (a+c)-(b+d) stays the same, and it also does when you convert it back to mods.
(mod m)
But what if I went and did
and
Then so.... (mod m) ???
Which is not true.
I've rearranged it so that there are two terms, x = 12 and y = 8+5-3. That's what happened in the proof. But my version doesn't work.
What the associativity rules and restrictions on this arithmetic?