Question

there is a spinner with 13 equal areas, numbered 1 through 13. if the spinner is spun one time, what is the probability that the result is a multiple of 3 and a multiple of 4?

Explanation:

Step1: Find LCM of 3 and 4

The least common multiple (LCM) of 3 and 4 is \( \text{LCM}(3,4) = 12 \), as 3 and 4 are coprime (their GCD is 1), so LCM is \( 3\times4 = 12 \).

Step2: Identify multiples in 1 - 13

We need numbers between 1 and 13 that are multiples of both 3 and 4, i.e., multiples of 12. In the range 1 to 13, the only multiple of 12 is 12 itself. So there is 1 favorable outcome.

Step3: Calculate probability

The total number of possible outcomes when spinning the spinner is 13 (since it's numbered 1 through 13). The probability \( P \) of an event is given by the formula \( P=\frac{\text{Number of favorable outcomes}}{\text{Total number of outcomes}} \). Here, the number of favorable outcomes is 1, and total outcomes is 13. So \( P = \frac{1}{13} \).

Answer:

\(\frac{1}{13}\)