Thread: linearity properties

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?

Originally Posted by DivideBy0

By the linearity property, . Re-arranging we get

How are you doing your rearrangement?


Then your example reads:
and

So

which is more or less what your initial statements imply.

-Dan

If and then for all .
In congruence notation we can prove this easily.
We know that and .
Multiply both side of the congruences by and respectively to get,
and .
Add them together,
.
Thus, .

Originally Posted by topsquark

How are you doing your rearrangement?


Then your example reads:
and

So

which is more or less what your initial statements imply.

-Dan

Sorry, that was supposed to be

As for my example, I think I got confused with the arithmetic somehow...