Prove that a/b, a/c then a/(b+c) - Thank you?

Question

Prove that a/b, a/c then a/(b+c) for all integers a/b, and also if a/b then a/bc for every interger c.
If d/x-y and d/x then d/y indirectly using other theories or consequences of other facts.

Answer 1

Ok I think you mean "a|b" i.e. "a divides b"; if so you definitely should use the vertical bar symbol to avoid confusion.

So if a|b, a|c that means there exist some integers p,q such that
b=pa , c = qa
Then (b+c)=pa+qa=(p+q)a
Noting that (p+q) must be an integer, therefore a|(b+c)

The other part is the same idea in disguise.
Set d=a , x=(b+c) and y=b, then (x-y) = c
etc.

Answer 2

When you say a devides b, then you can say b=xa.

From the given,

b=xa;c=ya
b+c=xa+ya=(x+y)a=za

z has to be an integer as x and y are integers.

means a/b+c

similarly

b=xa
bc=xca=za

x-y=ad as d devides x-y
x=bd as d devides x

Take the first equation

x-y=ad

x/d -y/d = a

Replace x/d with b

b-y/d=a -> y/d=b-a -> y=(b-a)d -> y=zd

Answer 3

Your question is unintelligible as stated!