Given Information:
- f(x 2) = f(x) f(x 1) for all positive integers x
- f(11) = 91
- f(15) = 617

To Find:
f(10)

Solution:

Step 1:
Using the given information, let's find the values of f(12), f(13), and f(14).

Since f(x 2) = f(x) f(x 1) for all positive integers x, we can say that:
f(12) = f(2) f(1)
f(13) = f(3) f(2)
f(14) = f(4) f(3)

Step 2:
Using the above equations, let's express f(12), f(13), and f(14) in terms of f(1) and f(2).

f(12) = f(2) f(1)
f(13) = f(3) f(2)
f(14) = f(4) f(3)

Since f(x 2) = f(x) f(x 1), we can express f(2) in terms of f(1):
f(2) = f(1) f(1 1) = f(1)^2

Similarly, f(3) = f(2) f(2 1) = f(2)^2 = (f(1)^2)^2 = f(1)^4

And f(4) = f(3) f(3 1) = f(3)^2 = (f(1)^4)^2 = f(1)^8

Step 3:
Now, let's substitute the values of f(2), f(3), and f(4) in terms of f(1) into the equations of f(12), f(13), and f(14).

f(12) = f(2) f(1) = (f(1)^2) f(1) = f(1)^3
f(13) = f(3) f(2) = (f(1)^4) (f(1)^2) = f(1)^6
f(14) = f(4) f(3) = (f(1)^8) (f(1)^4) = f(1)^12

Step 4:
Using the given values of f(11) and f(15), we can write the following equations:

f(11) = f(10) f(1)
91 = f(10) f(1)

f(15) = f(14) f(1)
617 = f(1)^12 f(1)

Step 5:
Now, let's solve the above equations to find the value of f(1) and then f(10).

From the equation 91 = f(10) f(1), we can see that f(10) must be a factor of 91. The factors of 91 are 1, 7, 13, and 91.

From the equation 617 = f(1)^12 f(1), we can see that f(1