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Homework Help: Divisibility homework problem

  1. Aug 16, 2010 #1
    1. The problem statement, all variables and given/known data

    if a and b are odd integer, then 8 l (a2-b2)

    2. Relevant equations

    n/a

    3. The attempt at a solution

    if a=b, clearly, 8 l (a2-b2)
    if not,
    now, i'm not sure how to continue

    should i varies b, and make a fixed, then varies a, and make b fixed,
    is that really the way to show, for all odd integer a and b?
     
  2. jcsd
  3. Aug 16, 2010 #2

    HallsofIvy

    User Avatar
    Science Advisor

    Re: divisibility

    [itex]a^2- b^2= (a+ b)(a- b)[/itex]

    If a and b are both odd we can write a= 2n+1 and b= 2m+ 1 for some integers m and n. The a+ b= 2(m+ n)+ 2= 2(m+n+1) is even and a- b= 2(m-n) is also even.

    That would be enough to show that [itex]a^2- b^2[/itex] is divisible by 4 but not enough to show it is divisible by 8.

    Of course, if a- b were divisible by 4 itself, then since a+ b is even, [itex]a^2- b^2= (a- b)(a+ b)[/itex] would be divisible by 8.

    Suppose a- b= 2(m- n) were not divisible by 4. That means that m- n must be an odd number: m- n= 2k+ 1 so that m= n+ 2k+ 1 and then m+ n= n+ 2k+ 1= 2n+ 2k+ 1= 2(m+k)+ 1, an odd number. But them m+n+ 1= 2(m+k)+ 1+ 1= 2(m+ k)+ 2= 2(m+k+1), and even number.

    That is, one of a- b and a+ b is even and the other a multiple of 4 so that their product, [itex]a^2- b^2[/itex] is a multiple of 8.
     
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