anonymous
  • anonymous
Can anyone prove the following? If each of a and b is in ℤ+ and a I b then a <= b
Mathematics
katieb
  • katieb
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KingGeorge
  • KingGeorge
What does it mean that a|b?
KingGeorge
  • KingGeorge
I.e., what's the definition of a|b?
anonymous
  • anonymous
means that there is an integer c such that b=a*c =====>>>> a | b

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KingGeorge
  • KingGeorge
Precisely, and both a,b are in \(\mathbb{Z}^+\). Now, assume that \(a>b\), and work towards a contradiction.
KingGeorge
  • KingGeorge
I would then break it up into two cases, and show why these are both impossible. (also, why is \(c\neq 0\)?) Case 1: \(c<0\) Case 2: \(c>0\)
anonymous
  • anonymous
contradiction when a>b?
KingGeorge
  • KingGeorge
Right. So assume a>b, and then divide into the two cases where c<0 and c>0. Reach a contradiction with both cases, so a<=b.
anonymous
  • anonymous
i'm sorry, i've never been good with contradictions. So like if c>0, then a would HAVE to be < b since (b/c) would result in a fraction?
anonymous
  • anonymous
therefore making a smaller than b?
KingGeorge
  • KingGeorge
Let's suppose \(c>0\). Then, \(c\ge 1\) since \(c\in\mathbb{Z}\). Hence, \[ac=b\ge a\]Since we assumed \(b
anonymous
  • anonymous
ohhh ok. Gotcha. Sorry for the bother
KingGeorge
  • KingGeorge
no problem.

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