Ans.

(64)

Given m and n are 2-digit numbers (m < n).

Such that gcd (m, n) = 6.

Let m = 6a, n = 6b, where a and b are coprime integers a < b.

$\Rightarrow 10\le m\le 99, 10\le n\le 99$.

$\Rightarrow 2\le a\le 16, 2\le b\le 16$.

Now, if a = 2, then b = 3, 5, 7, 9, 11, 13, 15 = 7

a = 3, then b = 4, 5, 7, 8, 10, 11, 13, 14, 16 = 9

a = 4, then b = 5, 7, 9, 11, 13, 15 = 6

a = 5, then b = 6, 7, 8, 9, 11, 12, 13, 14, 16 = 9

a = 6, then b = 7, 11, 13 = 3

a = 7, then b = 8, 9, 10, 11, 12, 13, 15, 16 = 8

a = 8, then b = 9, 11, 13, 15 = 4

a = 9, then b = 10, 11, 13, 14, 16 = 5

a = 10, then b = 11, 13 = 2

a = 11, then b = 12, 13, 14, 15, 16 = 5

a = 12, then b = 13 = 1

a = 13, then b  = 14, 15, 16 = 3

a = 14, then b = 15 = 1

a = 15, then b = 16 = 1

$\therefore$  Total possible number of ordered pairs = 64.