geometry

Since angles A and B are complementary, their measures sum up to 90 degrees. Let's denote the measure of angle B as $x$ degrees. Then, the measure of angle A is $90 - x$ degrees.

Given that angle A is a multiple of angle B, we can express angle A as $kx$, where $k$ is a positive integer.

So, we have:

$$kx = 90 - x$$

Solving for $x$, we get:

$$kx + x = 90$$

$$x(k + 1) = 90$$

$$x = \frac{90}{k + 1}$$

Now, since $x$ must be a positive integer and a divisor of 90, let's list down the possible values of $x$ and then find corresponding values of $k$:

1. If $k + 1 = 1$, then $x = 90$, but $x$ cannot be 90 as it's an integer less than 90.

2. If $k + 1 = 2$, then $x = 45$, which satisfies the conditions.

3. If $k + 1 = 3$, then $x = 30$, which also satisfies the conditions.

4. If $k + 1 = 5$, then $x = 18$, which satisfies the conditions.

5. If $k + 1 = 9$, then $x = 10$, which satisfies the conditions.

6. If $k + 1 = 10$, then $x = 9$, but 9 is not a divisor of 90.

7. If $k + 1 = 15$, then $x = 6$, which satisfies the conditions.

8. If $k + 1 = 18$, then $x = 5$, which satisfies the conditions.

9. If $k + 1 = 30$, then $x = 3$, which satisfies the conditions.

10. If $k + 1 = 45$, then $x = 2$, which satisfies the conditions.

11. If $k + 1 = 90$, then $x = 1$, but 1 is not a divisor of 90.

Therefore, the possible measures of angle A, expressed as multiples of angle B, are 2, 3, 5, 9, 15, 18, 30, 45. So, there are 8 possible measures for angle A.