Tuesday, 3 July 2012

Total Solutions

Find all possible non negative integer solutions of x,y. Assume k is a constant

1/x + 1/y = 1/k

3 comments:

  1. find any p,q s.t. k=p*q and gcd(p,q)=1... this can be done using fundamental theorem of arithmetic... k is uniquely representable by its prime factorization and p&q are partition of prime factorization of k i.e. p,q have different primes as their divisor and hence no common divisor.... now the solution is symmetric about x,y and the solution is x =(k+p), y = k*(k/p + 1) .... (note k/p = q, an integer)...

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  2. ^in the above argument, p can be any divisor of k, it need not be necessary that gcd(p,q)=1 and the solution remains the same....

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  3. @rik - The question is too find the total number of solutions rather than a particular solution

    Soln----

    1/x + 1/y = 1/k
    On simplifying we get
    (x-k)(y-k) = k^2
    Now, the total solutions would be nothing but the total factors of k^2 which can be easily find out.

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