Questions about the basic properties of Integers

[ninjagod123]

I am starting Number Theory this semester. My professor hands out notes but there is no textbook for the class. So hopefully you guys can help me with these seemingly easy problems.

Z = {...,-5,-4,-3,-2,-1,0,1,2,3,4,5,...}
Z is used to denote the set of integers

1) Show that if a is an element of Z and 0< a, then 1<=a

2) Let a and b be integers. Let us say that a divides b if there is an integer c such that b = ac. Show that if b>0 and a divides b then a<=b.

All we have learned so far are basic arithmetic properties, the Well-Ordering Principle, and the Induction Principle.

 

[wofsy]

I am starting Number Theory this semester. My professor hands out notes but there is no textbook for the class. So hopefully you guys can help me with these seemingly easy problems.

Z = {...,-5,-4,-3,-2,-1,0,1,2,3,4,5,...}
Z is used to denote the set of integers

1) Show that if a is an element of Z and 0< a, then 1<=a

2) Let a and b be integers. Let us say that a divides b if there is an integer c such that b = ac. Show that if b>0 and a divides b then a<=b.

All we have learned so far are basic arithmetic properties, the Well-Ordering Principle, and the Induction Principle.

I am not sure what you are allowed to use but it seems like you have some ordering on the integers that gives you an idea 0f < and <=. What is that?

 

[ninjagod123]

I am not sure what you are allowed to use but it seems like you have some ordering on the integers that gives you an idea 0f < and <=. What is that?

Hey sorry I don't understand. In class, some students attempted solutions but the solutions didn't satisfy the professor.

 

[wofsy]

Hey sorry I don't understand. In class, some students attempted solutions but the solutions didn't satisfy the professor.

what definition of < are you using?

The reason I ask is that I don't see what allows you to say that any number is greater or less than any other. Why is 3 < 12?

 

[ninjagod123]

what definition of < are you using?

The reason I ask is that I don't see what allows you to say that any number is greater or less than any other. Why is 3 < 12?

I don't know what kind of definitions there are. But since this was the first day, and the professor didn't say anything, I suppose we use the general definitions? I guess that's too vauge.

 

[ramsey2879]

I don't know what kind of definitions there are. But since this was the first day, and the professor didn't say anything, I suppose we use the general definitions? I guess that's too vauge.

What wofsy wants is the answer to the question
If a<b how does a-b relate to 0? The general definition of "<" will do fine.