An integer $N$ which satisfies exactly three of the four following conditions is called [i]two-good[/i].
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(I) $N$ is divisible by $2$
(II) $N$ is divisible by $4$
(III) $N$ is divisible by $8$
(IV) $N$ is divisible by $16$
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How many integers between $1$ and $100$, inclusive, are [i]two-good[/i]?

$\textbf{(A) }6\qquad\textbf{(B) }7\qquad\textbf{(C) }8\qquad\textbf{(D) }9\qquad\textbf{(E) }10$
To solve the problem, we need to identify integers \( N \) between 1 and 100 that satisfy exactly three of the four given conditions:
1. \( N \) is divisible by 2.
2. \( N \) is divisible by 4.
3. \( N \) is divisible by 8.
4. \( N \) is divisible by 16.

We will analyze each condition and determine which integers satisfy exactly three of these conditions.

1. **Condition Analysis:**
   - If \( N \) is divisible by 16, it is also divisible by 8, 4, and 2. Therefore, it satisfies all four conditions, which does not meet the requirement of being two-good.
   - If \( N \) is divisible by 8 but not by 16, it satisfies conditions (I), (II), and (III) but not (IV). This meets the requirement of being two-good.
   - If \( N \) is divisible by 4 but not by 8, it satisfies conditions (I) and (II) but not (III) and (IV). This does not meet the requirement of being two-good.
   - If \( N \) is divisible by 2 but not by 4, it satisfies condition (I) but not (II), (III), and (IV). This does not meet the requirement of being two-good.

2. **Finding Two-Good Numbers:**
   - Numbers divisible by 8 but not by 16: These numbers are of the form \( 8k \) where \( k \) is an odd integer.
   - We need to find such numbers between 1 and 100.

3. **Calculation:**
   - The smallest number divisible by 8 is 8.
   - The largest number divisible by 8 within 100 is 96.
   - The sequence of numbers divisible by 8 is: \( 8, 16, 24, 32, 40, 48, 56, 64, 72, 80, 88, 96 \).
   - We need to exclude numbers that are divisible by 16: \( 16, 32, 48, 64, 80, 96 \).

4. **Remaining Numbers:**
   - The numbers that are divisible by 8 but not by 16 are: \( 8, 24, 40, 56, 72, 88 \).

5. **Count:**
   - There are 6 such numbers.

The final answer is \(\boxed{6}\).