4
$\begingroup$

Suppose the quotient of two odd integers is an integer. Make and prove a conjecture about whether the quotient is either even or odd.

If you had an even (2n) divided by and odd (2m+1), it wont work. so my odd integers would be a= 2n+1 and b=2m+1

a/b = c = (2n+1)/(2m+1) which is also odd so c = (2w + 1)

c|a, so a = [(2n+1)/(2m+1)] * k for some integer k  

????? or I have

Let a = 2n +1 and b = 2m + 1. From the definition a/b = c and c|a, then we get a/b = c . Thus a = b*c = (2m+1)(c) = 4mc + c = c(4m + 1). Then, we have an equation that is (c)*(odd) making the final result odd.

examples: 9/3 = 3 21/7 = 3 81/9 = 9 49/7 = 7 35/5 = 7

  • 7
    Did you try some examples?2012-09-09
  • 1
    If this is HW, please tag it as such. Additionally, how would you expand an odd integer such that you can use the distributive rule?2012-09-09
  • 11
    Observe that if $a/b=c$ is an integer, then $c$ divides $a$. Can an even integer divide an odd one?2012-09-09
  • 1
    Sorry I am new to this website, It is a homework question. I am not sure how to even start this. I am very new to the solving proofs to begin with.2012-09-09
  • 1
    The way to start making a conjecture is to try some examples, as Robert Israel said. Have you tried some examples?2012-09-09
  • 0
    To complement what Dennis Gulko said, when attempting a proof, it helps to remember the definitions of keys concepts involved. In your case namely: what it means $a$ divides $b$ and the definition of odd and even numbers. Then think about the comment of Mr Gulko.2012-09-09
  • 0
    I have tried a few examples like 9/3 = 3 but 6/3 = 2 so i'm confused. Am I doing this right?2012-09-09
  • 0
    @Christene: $6$ isn't odd, so that has no bearing on what your conjecture should be.2012-09-09
  • 0
    Notice that $6$ is not odd. Keep trying with odd numbers only...2012-09-09
  • 0
    No, you wrote "quotient of two odd integers" but $6$ is not odd, it is even2012-09-09
  • 1
    Oh, sorry i didn't even realize it. 21/3 = 72012-09-09
  • 0
    Yes, keep trying and do not limit yourself to $3$ as denominator. Try other numbers. Then think about what Dennis Gulko said. After come back to edit your post with regard to your progress.2012-09-09
  • 1
    Correct me if I'm wrong but its more that I'm proving why it can't be even?2012-09-09
  • 0
    How does it look now? I'm not sure where to go from that.2012-09-09

2 Answers 2