+121 The measures of angles A and B are both positive, integer numbers of degrees. The measure of angle A is a multiple of the measure of angle B, and angles A and B are complementary angles. How many measures are possible for angle A?
+1768 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.
