To solve the problem, we first need to determine the total number of possible three-digit numbers and then find how many of those are divisible by 7. Finally, we calculate the probability.

Step 1: Total possible numbers

The number has the form 3*^, where $*$ and ^ represent digits from 0 to 9.

• The hundreds digit is fixed at $3$.
• The tens digit ($*$) can take any value from $0$ to $9$, so there are $10$ choices.
• The units digit (^) can also take any value from $0$ to $9$, giving another $10$ choices.

Thus, the total number of possible numbers is: $10 \times 10 = 100$

Step 2: Find numbers divisible by 7

To find how many numbers of the form 3*^ are divisible by 7:

1. The range of numbers is from $300$ to $399$, inclusive.
2. Determine the smallest number in this range divisible by 7: $300 \div 7 \approx 42.857 \quad \text{(round up to 43)}$ The smallest multiple of 7 is: $43 \times 7 = 301$
3. Determine the largest number in this range divisible by 7: $399 \div 7 \approx 57 \quad \text{(round down to 57)}$ The largest multiple of 7 is: $57 \times 7 = 399$

Now calculate the total number of multiples of 7 between 301 and 399, inclusive: $57 - 43 + 1 = 15$

Step 3: Calculate the probability

The probability is the ratio of favorable outcomes (numbers divisible by 7) to total outcomes: $\text{Probability} = \frac{15}{100} = 0.15$

Final Answer:

The probability that the number 3*^ is divisible by 7 is: $\boxed{0.15 \text{ or } 15\%}$

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Would you like a detailed breakdown of the steps or further clarifications?

Related Questions:

1. How do you verify divisibility by 7 for a number?
2. Can this method be generalized for divisors other than 7?
3. How does the range affect the calculation of probabilities?
4. What happens if $*$ and ^ are not restricted to single digits?
5. How can this problem be modeled using modular arithmetic?

Tip:

When working with divisibility problems, modular arithmetic is a powerful tool