Find the sum of all prime numbers between 58 and 68.

Introduction / Context:
This question tests basic knowledge of prime numbers and the ability to identify primes within a small interval. After locating each prime number between 58 and 68, you must add them to obtain the total sum. This type of question is common in exams to quickly check understanding of primality and simple addition.

Given Data / Assumptions:

• The interval is between 58 and 68, inclusive of the end points if they turn out to be prime.
• We must identify which numbers in this range are prime.
• Then we must compute the sum of those primes.
• Prime numbers are positive integers greater than 1 with exactly two distinct positive divisors, 1 and the number itself.

Concept / Approach:
The approach is to examine each integer from 58 through 68 and test whether it is divisible by any prime smaller than or equal to its square root. In this small range, we mainly test divisibility by 2, 3, 5, and 7. Once primes are identified, we simply add them. Because the range is narrow, this process is straightforward and does not require advanced techniques.

Step-by-Step Solution:
Step 1: List all integers from 58 to 68: 58, 59, 60, 61, 62, 63, 64, 65, 66, 67, 68. Step 2: Check 58. It is even, so it is divisible by 2 and therefore not prime. Step 3: Check 59. It is not divisible by 2, 3, 5, or 7, so 59 is prime. Step 4: Check 60. It is even and divisible by 2, so it is not prime. Step 5: Check 61. It is not divisible by 2, 3, 5, or 7, so 61 is prime. Step 6: Check 62. It is even, so it is not prime. Step 7: Check 63. It is divisible by 3 and 7, so it is not prime. Step 8: Check 64. It is 8 squared, so it is clearly composite. Step 9: Check 65. It ends with 5, so it is divisible by 5 and not prime. Step 10: Check 66. It is even and divisible by 2 and 3, so it is not prime. Step 11: Check 67. It is not divisible by 2, 3, 5, or 7, so 67 is prime. Step 12: Check 68. It is even and divisible by 2, so it is not prime. Step 13: The primes in this range are therefore 59, 61, and 67. Step 14: Compute the sum: 59 + 61 + 67 = 120 + 67 = 187.

Verification / Alternative check:
You can verify by rechecking divisibility of each candidate prime. For example, for 59, test division by 2, 3, 5, and 7, and confirm no exact division occurs. Similarly, check 61 and 67, noting that there is no divisor other than 1 and themselves. Recalculate the sum to ensure no arithmetic error: 59 + 61 = 120, then 120 + 67 = 187. Since all checks are consistent and no other primes exist in the range, 187 must be correct.

Why Other Options Are Wrong:
The other options 179, 178, 183, and 191 correspond to incorrect combinations or sums. They would arise if a composite number like 65 were mistakenly included or if one of the primes were omitted or miscalculated. For instance, including 65 by mistake and excluding 67 could produce incorrect totals. Only 187 matches the exact sum of the verified primes 59, 61, and 67.

Common Pitfalls:
A frequent error is to forget 59 or 67 as primes because they are less familiar than smaller primes. Another common pitfall is to miscalculate the sum or to treat 65 or 63 as primes without checking divisibility by 5 or 3. To avoid mistakes, always test suspected primes for divisibility by small primes up to their square root and perform addition carefully, possibly in stages, as shown. This systematic approach ensures accuracy.

Final Answer:
The sum of all prime numbers between 58 and 68 is 187.