" If "p,q" are prime positive integers,"

Let lim _( x to oo) n ^((1)/(2 )(1+(1 )/(n))). (1 ^(1) . 2 ^(2) . 3 ^(3)....n ^(n ))^((1)/(n ^(2)))=e^((-p)/(q)) where p and q are relative prime positive integers. Find the value of |p+q|.

If r,s,t are prime numbers and p,q are the positive integers such that LCM of p,q is r^(2)s^(4)t^(2) , then the number of ordered pairs (p,q) is:

If r,s,t are prime numbers and p,q are the positive integers such that LCM of p,q is r^2t^4s^2 , then find the number of ordered pair (p,q).

lf r, s, t are prime numbers and p, q are the positive integers such that their LCM of p,q is r^2 t^4 s^2, then the numbers of ordered pair of (p, q) is

lf r, s, t are prime numbers and p, q are the positive integers such that their LCM of p,q is r^2 t^4 s^2, then the numbers of ordered pair of (p, q) is

If p, q are positive integers, f is a function defined for positive numbers and attains only positive values such that f(xf(y))=x^p y^q , then prove that p^2=q .

If p, q are positive integers, f is a function defined for positive numbers and attains only positive values such that f(xf(y))=x^p y^q , then prove that p^2=q .