The number of pairs (m,n) of positive integers satisfying `1lt=mltnlt=30` have to be determined so that a real number x exists that allows `sin(mx) + sin(nx) = 2` .

For the sine function, the maximum value of sin x is 1 when `x = (2*N*pi+pi/2)` for an integer N.

If `sin(mx) + sin(nx) = 2` , then both `sin (mx)` and `sin (nx)` should be equal to 1.

As m and n are positive integers, `x = pi/2` .

For `1lt=mltnlt=30` , the possible pairs of m and n are

(1, 5), (1, 9), (1,13), (1,17), (1,21), (1,25), (1,29)

(5, 9), (5,13), (5,17), (5,21), (5,25), (5,29)

(9,13), (9,17), (9,21), (9,25), (9,29)

(13,17), (13,21), (13,25), (13,29)

(17,21), (17,25), (17,29)

(21,25), (21,29)

(25,29).

The total number of pairs that satisfy the given conditions is 28.