A skeet shooting competition awards prizes for each round as follows : the first - place winner receives 11 points, the second - place winner receives 7 points, the third - place finisher receives 5 points, and the fourth - place finisher receives 2 points. No other prizes are awardded Johan completes in several rounds of the skeet shooting competition and receives points every time he completes. If the product of all of hte points he receives equal 84,700, in how many rounds does he participate?

Answer

To solve the problem, we need to find out how many rounds Johan participated in based on the product of the points he received. The points awarded for each place are: 11 points for first place, 7 points for second place, 5 points for third place, and 2 points for fourth place.

Given that the product of all the points Johan receives equals 84,700, we will perform the prime factorization of 84,700 to determine how many rounds he participated in.

### Step-by-step Solution:

1. **Start with the number 84,700**:
- We will perform prime factorization on this number.

2. **Divide by 2** (the smallest prime number):
- \( 84700 \div 2 = 42350 \)
- \( 42350 \div 2 = 21175 \)

3. **Next, divide by 5** (since the last digit of 21175 is 5):
- \( 21175 \div 5 = 4235 \)
- \( 4235 \div 5 = 847 \)

4. **Now, check 847 for divisibility**:
- 847 is not even, so we cannot divide by 2. The last digit is not 0 or 5, so we cannot divide by 5. We will check for 7 and 11.
- \( 847 \div 7 = 121 \)

5. **Now factor 121**:
- \( 121 = 11 \times 11 \) (or \( 11^2 \))

6. **Now we can summarize the prime factorization**:
- \( 84,700 = 2^2 \times 5^2 \times 7^1 \times 11^2 \)

7. **Count the total number of factors**:
- From the prime factorization:
- \( 2^2 \) contributes 2 factors (2 rounds)
- \( 5^2 \) contributes 2 factors (2 rounds)
- \( 7^1 \) contributes 1 factor (1 round)
- \( 11^2 \) contributes 2 factors (2 rounds)

- Total factors = \( 2 + 2 + 1 + 2 = 7 \)

8. **Conclusion**:
- Johan participated in **7 rounds** of skeet shooting.