Question

This isn't a discrete math question but I couldn't decide which section to post my question under so hopefully someone here can help me..

Suppose m is a positive integer. Prove that for all integers a,b,c and d, if a=c [where "=" means congruent] (mod m) and b=d (mod m) then ab=cd (mod m).

Help would be most appreciated. Thank you.

Answer 1

ok i just fiddled around. Cant promise Im correct.

Suppose m is a positive integer and a,b ,c and d are integers such that a = c mod m and b= d mod m. Using the definition of mod we have:
a=c+km where k is an integer and
b=d+lm where l is an integer.
So (a*b)=(c+km)(d+lm)
=cd+klm^2+ckm+dlm
= cd+(klm+ck+dl) m
=cd+jm, where j is an integer equaling (klm+ck+dl)
= cd mod m, Using the definition of mod
Hence we conclude that for integers a,b,c and d if a=c mod m and b=d mod m then ab=cd mod m

Answer 2

thats about the gist of it yep. might want to include that j is an integer by closure property of integers

Answer 3

Oh yayyy!!!

Answer 4

Ahh okay that makes sense. I was trying to multiply (a-c) and (b-d) together but I had a feeling that wasn't correct. Thank you to you both!

Answer 5

:) yw