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)...

@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.

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)...

ReplyDelete^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....

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

ReplyDeleteSoln----

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.