Fifty cards, numbered 1-50, are placed in a box. One card is randomly selected. What is the probability that the number on the card is prime and is a multiple of 7? Express your answer as a common fraction.
The only multiple of 7 that is prime is 7. Any other multiple of 7 has at least three positive divisors: 1, 7, and itself. Therefore, the probability that the selected number is both prime and a multiple of 7 is $\boxed{\frac{1}{50}}$.
\frac{1}{50}
Level 2
Number Theory
Problem: Fifty cards, numbered 1-50, are placed in a box. One card is randomly selected. What is the probability that the number on the card is prime and is a multiple of 7? Express your answer as a common fraction.
Solution:<|START_OF_LATENT|><|PRIOR_PREFIX|>To find the probability that a randomly selected card is both prime and a multiple of 7, we first identify the characteristics of prime numbers and multiples of 7. A prime number is defined as a natural number greater than 1 that has no positive divisors other than 1 and itself. A multiple of 7 is any number that can be expressed as 7 times an integer.

Next, we list the multiples of 7 within the range of 1 to 50. These multiples are: 7, 14, 21, 28, 35, 42, and 49. This gives us a total of 7 multiples of 7.

Now, we need to check which of these multiples are prime. We evaluate each multiple:
- 7 is prime (divisors: 1, 7).
- 14 is not prime (divisors: 1, 2, 7, 14).
- 21 is not prime (divisors: 1, 3, 7, 21).
- 28 is not prime (divisors: 1, 2, 4, 7, 14, 28).
- 35 is not prime (divisors: 1, 5, 7, 35).
- 42 is not prime (divisors: 1, 2, 3, 6, 7, 14, 21, 42).
- 49 is not prime (divisors: 1, 7, 49).

From this evaluation, the only multiple of 7 that is also prime is 7 itself. Therefore, there is only 1 card that meets the criteria of being both prime and a multiple of 7.

Next, we calculate the total number of possible outcomes, which is the total number of cards, 50. The probability of drawing a card that is both prime and a multiple of 7 is given by the ratio of favorable outcomes to total outcomes. This is calculated as follows:

Probability = (Number of favorable outcomes) / (Total number of outcomes) = 1 / 50.

Thus, the final answer is $\frac{1}{50}$.<|END_OF_LATENT|> The final answer is $\frac{1}{50}$.