What is the sum of all prime numbers between 60 and 80?

Introduction / Context:
This question focuses on identifying prime numbers in a small interval and then summing them. It tests both knowledge of basic primality checks and careful arithmetic when adding the identified primes together.

Given Data / Assumptions:

- We consider integers strictly between 60 and 80.

- We must identify which of these integers are prime numbers.

- Then we must add all these prime numbers to find their total sum.

Concept / Approach:
A prime number is a number greater than 1 with exactly two positive divisors: 1 and itself. To test whether a number in this range is prime, it is sufficient to check divisibility by smaller primes such as 2, 3, 5, 7, and 11. Once we list all primes from 61 to 79, we can add them together.

Step-by-Step Solution:
Step 1: List integers from 61 to 79. The numbers are 61, 62, 63, 64, 65, 66, 67, 68, 69, 70, 71, 72, 73, 74, 75, 76, 77, 78, 79. Step 2: Identify primes in this list. - 61: Not divisible by 2, 3, 5, or 7 → prime. - 67: Not divisible by 2, 3, 5, or 7 → prime. - 71: Not divisible by 2, 3, 5, or 7 → prime. - 73: Not divisible by 2, 3, 5, or 7 → prime. - 79: Not divisible by 2, 3, 5, or 7 → prime. Other numbers in the range are composite (for example, 63 is divisible by 3, 65 by 5, 69 by 3, 77 by 7, etc.). Step 3: Sum these primes. Sum = 61 + 67 + 71 + 73 + 79. First, 61 + 67 = 128. 128 + 71 = 199. 199 + 73 = 272. 272 + 79 = 351. So the sum is 351.

Verification / Alternative check:
Check quickly that no prime is missed: The only candidates in this range that are not obviously even or multiples of 5 are 61, 67, 71, 73, 79. All have been tested for divisibility by small primes, confirming their primality. Hence, the list is complete and the sum 351 is correct.

Why Other Options Are Wrong:
- 272, 284, 414: These numbers do not equal the sum of the prime numbers between 60 and 80. They likely arise from partial sums or misidentified primes.

Common Pitfalls:
Some learners may accidentally include composite numbers like 69 or 77, or forget one of the primes such as 79. Others miscalculate the sum. A systematic scan for divisibility and careful addition step by step prevents these errors.

Final Answer:
The sum of all prime numbers between 60 and 80 is 351.