1. ## Integer basic proofs

Hello,

Can anyone help me with any of the foolowing proofs

For any integers a,b,c,d
1. a | 0, 1 | a, a | a.
2. a | 1 if and only if a=+/-1.
3. If a | b and c | d, then ac | bd.
4. If a | b and b | c, then a | c.
5. a | b and b | a if and only if a=+/-b.
6. If a | b and b is not zero, then |a| < |b|.
7. a+b is integer, a.b is integer
Thank you

2. Originally Posted by Dili
Hello,

Can anyone help me with any of the foolowing proofs

For any integers a,b,c,d
1. a | 0, 1 | a, a | a.
2. a | 1 if and only if a=+/-1.
3. If a | b and c | d, then ac | bd.
4. If a | b and b | c, then a | c.
5. a | b and b | a if and only if a=+/-b.
6. If a | b and b is not zero, then |a| < |b|.
7. a+b is integer, a.b is integer
Thank you
Let me do the first ones.

Definition: Let $\displaystyle a,b\in \mathbb{Z}$ then we define $\displaystyle a|b$ iff $\displaystyle b=ac \mbox{ for some }c\in \mathbb{Z}$.

Theorem: Let $\displaystyle a\in \mathbb{Z}$ then $\displaystyle a|0$.

Proof: We need to show $\displaystyle 0=ac$ for some $\displaystyle c\in \mathbb{Z}$. So choose $\displaystyle c=0$.Q>E>D>

Theorem: Let $\displaystyle a\in \mathbb{Z}$ then $\displaystyle 1|a$.

Proof: We need to show $\displaystyle a=1c$ for some $\displaystyle c\in \mathbb{Z}$. So choose $\displaystyle c=a$.Q.E.D.

Theorem: Let $\displaystyle a\in \mathbb{Z}$ then $\displaystyle a|a$.

Proof: We need to show $\displaystyle a=ac$ for some $\displaystyle c\in \mathbb{Z}$. So chose $\displaystyle c=1$.Q.E.D.

Theorem: Let $\displaystyle a\in \mathbb{Z}$ and $\displaystyle a|1$ then $\displaystyle a=\pm 1$.

Proof: We need to find $\displaystyle 1=ac$ for some $\displaystyle c\in \mathbb{Z}$. If $\displaystyle |a|\geq 2$ then $\displaystyle |ac|>1$ which is impossible. If $\displaystyle a=0$ it is impossible. So the only possible case is $\displaystyle |a|=1$.Q.E.D.

3. Thank you.
I shall welcome any other proofs to the others especially the 7th one
ie. a+b and a.b are integers

4. Originally Posted by Dili
ie. a+b and a.b are integers
That has nothing to do with number theory. That is more of a Set Theory question. So I have no idea what you want with that one.

5. For the seventh one, isnt that the very definition of a field? Really I think its a closure axiom, so you dont need to prove it.

6. Originally Posted by tukeywilliams
For the seventh one, isnt that the very definition of a field?
The integers are not a field.

Really I think its a closure axiom, so you dont need to prove it.
That is not how this axiom thing works, i.e. you cannot simply say it is an axiom. Before you state something is closed, you need to actually show it is closed. You cannot just say that is an axiom, that makes no sense.

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This question (7) should just be avoided. It has nothing to do with number theory.