Find the positive integers p, q, r, s satisfying `tan(pi/24)= (sqrt p - sqrt q)(sqrt r -s )`

If p and q are positive integers, then which one of the following equations has p-sqrt(q) as one of its roots ?

If p + sqrt q = r + sqrts , then

A = [[p,q],[r,s]] where p, q, r and s are positive integers. If A is symmetric matrix, |A|=20 and p = q, then find how many values are possible for s.

If p,q are prime positive integers,prove that sqrt(p)+sqrt(q) is an irrational number.

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