To determine the probability that the result of spinning a spinner numbered 1 through 12 is a multiple of both 6 and 4, we first need to identify the common multiples of these two numbers.
The least common multiple (LCM) of 6 and 4 can be calculated as follows:
- The prime factorization of 6 is \(2 \times 3\).
- The prime factorization of 4 is \(2^2\).
To find the LCM, we take the highest power of each prime factor present in the two numbers:
\[ \text{LCM}(6, 4) = 2^2 \times 3^1 = 4 \times 3 = 12 \]
Now that we know the LCM, we can find the multiples of 12 in the range of numbers on the spinner (1 to 12). There is only one multiple of 12 in this range, which is 12 itself.
Next, the total number of outcomes when spinning the spinner is 12 (the numbers 1 through 12).
Thus, the probability \(P\) of the spinner landing on a number that is a multiple of both 6 and 4 is given by the ratio of the number of favorable outcomes to the total number of outcomes:
\[ P = \frac{\text{Number of favorable outcomes}}{\text{Total number of outcomes}} = \frac{1}{12} \]
Therefore, the answer is:
1/12