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Questions about the basic properties of Integers

  1. Jan 19, 2010 #1
    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.
  2. jcsd
  3. Jan 19, 2010 #2
    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?
  4. Jan 19, 2010 #3
    Hey sorry I don't understand. In class, some students attempted solutions but the solutions didn't satisfy the professor.
  5. Jan 19, 2010 #4
    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?
  6. Jan 19, 2010 #5
    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.
  7. Jan 21, 2010 #6
    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.
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