v (mod d) and w x (mod d), then u + w v + x (mod d). To prove that an if--then statement (p
q) is true, we assume that p is true and show that q is also
true.5a) The statement we are required to prove is: If u v (mod d) and w x (mod d), then u + w v + x (mod d).
To prove that an if--then statement (p q) is true, we assume that p is true and show that q is also
true.
Recall that to say that a b (mod d) is equivalent to saying:
1. a mod d = r and b mod d = r,
2. a - b = kd for some integer k, or
3. d | (a - b).
Use the second point from the list above to rewrite u v (mod d) and w x (mod d), as equations.
Proof Suppose that u, v, w, x and d are integers
such that u v (mod d)
and w x (mod d).
By the definition of congruence this is equivalent to saying that u - v = sd and w - x = td, for some integers s and t.
Adding these two equations we see that: (u - v) + (w - x) |
= |
sd + td |
(u + w) - (v + x) |
= |
(s + t)d |
Since s and t are both integers, we know that s + t is also an
integer. Hence, (u + w) - (v + x) = md, for some integer m.
By point 2 in the definition of congruence, this is equivalent to saying that
u + w v + x (mod d).