# Theorem regarding mod properties

1. Jun 2, 2005

### plusunim

There is a theorem regarding mod properties such that when a=a'(mod
c) and b=b'(mod c) then a*±b=a'±b' (mod c) right?

Now, 5=5 mod7. applying it to the above, 10=10 mod7, which is not
true since 3=10 mod7. Why? I'm confused.

Thanks

2. Jun 2, 2005

### matt grime

No 10 is congruent to 10 mod 7 and it is congruent to 3. You are mistaken only in thinking this is wrong. really you ought to be susing congruent signs, not equal signs. Any equivalence class has infintely many elements in it in modulo arithmetic.

3. Jun 2, 2005

### plusunim

howcome 10=10 mod7? I thought that you can have the same numbers on both sides only when the numbers are less than the mod. Am I wrong again?

-- and by the way, to type fast I use equal signs, I know it's not fully correct, please bear my momentaneous impatience --

4. Jun 2, 2005

### matt grime

Why would you only be able to have the same numbers on both sides if they were sufficiently small? x=y mod n means exactly that n divides x-y or equivalently that there is an integerk such that x=kn+y and that is all.

5. Jun 2, 2005

### Gokul43201

Staff Emeritus
As matt has said already, start from the definition of congruence. You are confusing a residue with a least positive residue (commonly refered to as a remainder).

Again, start from the beginning and work your way up.

6. Jun 4, 2005

### plusunim

yes I think so. I'll continue reading on. Thanks for the help :)