Mathematical proofs. number theory proofs?

Both of these questions have you looking for immediate consequences of the definition of the notion "x divides y". If you're struggling, I'd start with that definition.

"x divides y" means that you can find an integer h so that xh = y

So for your first problem, you know that a divides b and that c divides d. That means that you know you have two integers h and k so that ah = b and ck = d.

You are asked to prove that ac divides bd, meaning you have to show that there exists an integer x so that (ac)x = bd

Can you see how to use ah = b and ck = d to try and create a new equation in the form (ac)x = bd?

(Your second problem is very similar.)

there are extremely some undetermined ( now now no longer undefined ) mathematical elements mutually with : infinity - infinity infinity / infinity 0 / 0 0 ^ 0 0 * infinity as an social gathering : (0^2)^0 = 0^0 (0^3)^0 = 0^0 is that recommend that 2=3 ? So 0^0 is an undetermined quantity on the concern of that 0 might want to be considered by employing way of truth the smallest +ve quantity we are able to ever have or the numerous important -ve quantity we are able to ever have

1)
a|b, c|d
Let p and q be integers such that
ap = b, cq = d
(ac)(pq) = bd
Therefore, ac divides bd.

2)
ac|bc
Let p be an integer such that
acp = bc
Since c does not equal 0,
ap = b
Therefore, a divides b if and only if ac divides bc (c does not equal 0).