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Math Help - Proof validation

  1. #1
    Junior Member
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    Dec 2005
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    Proof validation

    I need to prove that if a,b are integers then a^2-b^2=101010 is not possible.

    My proof is this:
    Since 101010 can be divided with 5 which is prime number, then one of factors (a-b)(a+b) also must be 5.
    Lets assume that a-b=5 and a+b=5k.
    Because it must be (a-b)(a+b)=101010 then it must be a+b=20202.
    So, we have to solve system of equations in order to see whether a,b are integers.

    \begin{array}{l}<br />
 \left\{ \begin{array}{l}<br />
 a - b = 5 \\ <br />
 a + b = 20202 \\ <br />
 \end{array} \right. \\ <br />
  \\ <br />
 \left\{ \begin{array}{l}<br />
 a = 5 + b \\ <br />
 a = 20202 - b \\ <br />
 \end{array} \right. \\ <br />
  \\ <br />
 5 + b = 20202 - b \\ <br />
 2b = 20202 - 5 \\ <br />
 b = \frac{{20197}}{2} \\ <br />
 \end{array}<br />

    Since \frac{{20197}}{2} is not integer then b is not integer.

    Is this proof valid?
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  2. #2
    Grand Panjandrum
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    Quote Originally Posted by DenMac21
    I need to prove that if a,b are integers then a^2-b^2=101010 is not possible.

    My proof is this:
    Since 101010 can be divided with 5 which is prime number, then one of factors (a-b)(a+b) also must be 5.
    Lets assume that a-b=5 and a+b=5k.
    101010 divisible by 5 implies one of (a-b) and (a+b) is divisible by 5, not
    that one of them is 5. Also if one were 5 the other need not be divisible
    by 5.

    Because it must be (a-b)(a+b)=101010 then it must be a+b=20202.
    So, we have to solve system of equations in order to see whether a,b are integers.

    \begin{array}{l}<br />
 \left\{ \begin{array}{l}<br />
 a - b = 5 \\ <br />
 a + b = 20202 \\ <br />
 \end{array} \right. \\ <br />
  \\ <br />
 \left\{ \begin{array}{l}<br />
 a = 5 + b \\ <br />
 a = 20202 - b \\ <br />
 \end{array} \right. \\ <br />
  \\ <br />
 5 + b = 20202 - b \\ <br />
 2b = 20202 - 5 \\ <br />
 b = \frac{{20197}}{2} \\ <br />
 \end{array}<br />

    Since \frac{{20197}}{2} is not integer then b is not integer.

    Is this proof valid?
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