Some Simple Division Lemmas

In this blog, I want to outline some simple implications of the division which are very useful in basic mathematical reasoning:

Lemma 1:

If:

(a) c = a + b
(b) d divides a
(c) d divides b

Then:

d divides c

(1) By assumption (b) and (c), we know that there exists a',b' such that:

a = a'd
b = b'd

(2) So,

c = (a'd) + (b'd) = d(a' + b')

Lemma 2:

If

(a) c = a + b
(b) d divides c
(c) d divides a

Then

d divides b

(1) c = a + b → b = c -a = c + (-a)

(2) This results follows from Lemma 1 above.