Theorem regarding mod properties

[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

 

[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 infinitely many elements in it in modulo arithmetic.

 

[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 --

Thanks for replying anyway

 

[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.

 

[Gokul43201]

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.

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

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

 

[plusunim]

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