Compute the number of integers between $1$ and $100$, inclusive, that have an odd number of factors. Note that $1$ and $4$ are the first two such numbers.
To determine the number of integers between \(1\) and \(100\) that have an odd number of factors, we need to understand the conditions under which a number has an odd number of factors.

1. **Understanding Factors**:
   - A number \(n\) has an odd number of factors if and only if \(n\) is a perfect square. This is because factors generally come in pairs \((d, \frac{n}{d})\), except when \(d = \frac{n}{d}\), which happens only when \(n\) is a perfect square.

2. **Identifying Perfect Squares**:
   - We need to find all perfect squares between \(1\) and \(100\). These are the squares of integers from \(1\) to \(\lfloor \sqrt{100} \rfloor = 10\).

3. **Listing Perfect Squares**:
   - The perfect squares between \(1\) and \(100\) are:
     \[
     1^2 = 1, \quad 2^2 = 4, \quad 3^2 = 9, \quad 4^2 = 16, \quad 5^2 = 25, \quad 6^2 = 36, \quad 7^2 = 49, \quad 8^2 = 64, \quad 9^2 = 81, \quad 10^2 = 100
     \]

4. **Counting the Perfect Squares**:
   - There are \(10\) perfect squares in the list above.

Therefore, the number of integers between \(1\) and \(100\) that have an odd number of factors is \(\boxed{10}\).