Powers of Two Divisibility

$0.25 = 2^{-2} < 2^x < 200 \Rightarrow -1 \le x \le 7$ for integer $x$. Possible integer values of $x$: $\{-1, 0, 1, 2, 3, 4, 5, 6, 7\}$. Test values:

• $x = -1$: $2^{-1} + 2 = 2.5$ (not integer)
• $x = 0$: $2^0 + 2 = 3$ (divisible by 3) - Yes
• $x = 1$: $2^1 + 2 = 4$ (divisible by 4) - Yes
• $x = 2$: $2^2 + 2 = 6$ (divisible by 3) - Yes
• $x = 3$: $2^3 + 2 = 10$ (not div by 3 or 4)
• $x = 4$: $2^4 + 2 = 18$ (divisible by 3) - Yes
• $x = 5$: $2^5 + 2 = 34$ (not div by 3 or 4)
• $x = 6$: $2^6 + 2 = 66$ (divisible by 3) - Yes
• $x = 7$: $2^7 + 2 = 130$ (not div by 3 or 4) Satisfying values: $x \in \{0, 1, 2, 4, 6\}$. Total = 5.