Finding the Average of Prime Numbers Between 68 and 96
The question asks us to calculate the average value of all the prime numbers that lie in the range strictly between 68 and 96.
Step 1: Identify Prime Numbers Between 68 and 96
First, we need to list all the numbers between 68 and 96. Then, we identify which of these are prime numbers. A prime number is a natural number greater than 1 that has only two divisors: 1 and itself.
- Numbers to check: 69, 70, 71, 72, 73, 74, 75, 76, 77, 78, 79, 80, 81, 82, 83, 84, 85, 86, 87, 88, 89, 90, 91, 92, 93, 94, 95.
- Checking for primality:
- 69 is divisible by 3.
- 70 is divisible by 2, 5, 7, 10.
- 71 is prime (only divisible by 1 and 71).
- 72 is divisible by 2.
- 73 is prime (only divisible by 1 and 73).
- 74 is divisible by 2.
- 75 is divisible by 3, 5.
- 76 is divisible by 2.
- 77 is divisible by 7, 11.
- 78 is divisible by 2.
- 79 is prime (only divisible by 1 and 79).
- 80 is divisible by 2, 5.
- 81 is divisible by 3, 9.
- 82 is divisible by 2.
- 83 is prime (only divisible by 1 and 83).
- 84 is divisible by 2.
- 85 is divisible by 5.
- 86 is divisible by 2.
- 87 is divisible by 3.
- 88 is divisible by 2.
- 89 is prime (only divisible by 1 and 89).
- 90 is divisible by 2, 3, 5.
- 91 is divisible by 7, 13.
- 92 is divisible by 2.
- 93 is divisible by 3.
- 94 is divisible by 2.
- 95 is divisible by 5.
- The prime numbers lying between 68 and 96 are: 71, 73, 79, 83, 89.
Step 2: Calculate the Sum of the Prime Numbers
Next, we add all the prime numbers we found.
Sum = $71 + 73 + 79 + 83 + 89$
Sum = $395$
Step 3: Count the Number of Prime Numbers Found
We count how many prime numbers are in our list.
Count = 5
Step 4: Calculate the Average
The average is calculated by dividing the sum of the numbers by the count of the numbers.
Average = $\frac{\text{Sum of Prime Numbers}}{\text{Count of Prime Numbers}}$
Average = $\frac{395}{5}$
Average = $79$
Conclusion
Therefore, the average of the prime numbers lying between 68 and 96 is 79.
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