Sample Question 7:

A regular 15-gon has $L$ lines of symmetry, and the smallest positive angle for which it has rotational symmetry is $R$ degrees. What is $L+R$ ?

$\textbf{(A)}\; 24 \qquad\textbf{(B)}\; 27 \qquad\textbf{(C)}\; 32 \qquad\textbf{(D)}\; 39 \qquad\textbf{(E)}\; 54$

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Answer Keys

Question 7: D

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Solutions

Question 7
Step 1: Recognize that a regular 15-gon has as many lines of symmetry as it has vertices. Thus, $L=15$.

Step 2: Consider that the smallest angle for rotational symmetry is the angle that turns one side of the polygon into an adjacent side. In other words, the entire 360 degrees is equally divided among the sides of polygon. Therefore, $R = 360^\circ / 15 = 24^\circ$.

Step 3: To find $L+R$, simply sum the determined values of $L$ and $R$.

Step 4: Using the calculated values from steps 1 and 2, add $L$ to $R$ to get $L + R = 15 + 24 = 39$.

Step 5: Therefore, the answer is $\boxed{\textbf{(D)} \, 39}$.