Problem

Let

\log_a N = \alpha + \beta

where

\alpha

is integer and

\beta \in [0, 1)

. Then, On the basis of above information, answer the following questions. If

N_1

is number of integers when

a = 2

and

\alpha = 2

and

N_2

is number of integers when

\alpha = 1

and

a = 3

, then the minimum value of

(N_1 \sec^2 \theta + N_2 \csc^2 \theta)

10 + 4\sqrt{6}

10 + \sqrt{6}

c. 10 d. 100