1. ## Divisibility of 0

Hey all, any help on the following proofs would be appreciated:

(i) 0 is divisible by every integer

(ii) If m is an integer not equal to 0, then m is not divisible by 0.

2. Originally Posted by jstarks44444
Hey all, any help on the following proofs would be appreciated:

(i) 0 is divisible by every integer

(ii) If m is an integer not equal to 0, then m is not divisible by 0.
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For (i) use the fact that 0 = 0*a, for any integer a

For (ii), suppose not. Suppose that a/0 = b for a a non-zero integer and b some real number. try to find a contradiction, using the axiom 0 =! 1

3. Originally Posted by jstarks44444
Hey all, any help on the following proofs would be appreciated:

(i) 0 is divisible by every integer

(ii) If m is an integer not equal to 0, then m is not divisible by 0.
(i) For all $n\in\mathbb{Z}, 0=0\cdot n\implies n\mid 0$.

(ii) Suppose $0\mid m$. Then $\exists\,k\in\mathbb{Z}: m = k\cdot 0$. So what can be said about $m$?

4. This is another example of a very basic principle- proofs follow from the precise words of definitions.

"m is divisible by n" means that there exist some integer, k, such that m= nk.

If m= 0, is there an integer, k, such that nk= 0?

If m is NOT 0, does there exist an integer, k, such that 0k= m?