Find the total number of positive factors of 9321.

Introduction / Context:
This question checks your ability to factorise a composite number into primes and then use the prime factorisation to count the total number of its positive divisors. This is a standard number theory technique useful in many aptitude and competitive exams. The key is to express the number as a product of powers of primes and then apply a simple rule for counting factors.

Given Data / Assumptions:

• The given number is 9321.
• We are interested in positive factors only.
• We assume basic knowledge of divisibility tests and prime numbers.
• The factor counting formula for a number in the form p^a * q^b * r^c is (a + 1)(b + 1)(c + 1).

Concept / Approach:
First, we check whether 9321 is divisible by small primes such as 2, 3, 5, 7, 11, 13 and so on. Divisibility by 3 is quickly tested using the sum of digits. After we find one prime factor, we divide the number by that factor and continue to factor the quotient. Once we have the full prime factorisation, we apply the exponent based rule to count all positive factors. This method is far more efficient than listing every divisor by trial.

Step-by-Step Solution:
Given number N = 9321. Check divisibility by 3: sum of digits = 9 + 3 + 2 + 1 = 15, which is divisible by 3, so 9321 is divisible by 3. Compute 9321 / 3 = 3107. Now factor 3107. Check divisibility by small primes: 3107 is not divisible by 3, 5 or 7. Test 13: 13 * 239 = 3107, so 3107 = 13 * 239. Check 239 for primality. It is not divisible by 2, 3, 5, 7, 11 or 13, so 239 is prime. Therefore, prime factorisation of 9321 is 3^1 * 13^1 * 239^1. Number of positive factors = (1 + 1)(1 + 1)(1 + 1) = 2 * 2 * 2 = 8.

Verification / Alternative check:
We can list the factors explicitly to verify. From primes 3, 13 and 239, every factor is a product of a subset of these primes. The factors are 1, 3, 13, 39, 239, 3 * 239 = 717, 13 * 239 = 3107 and 3 * 13 * 239 = 9321. Counting them gives exactly 8 factors, matching the result from the exponent formula. This confirms that the factorisation and counting are both correct.

Why Other Options Are Wrong:

• 4 and 5: These would correspond to too few factors and would imply that the number has fewer distinct prime factors or lower exponents than it actually has.
• 6 and 7: These are closer but still do not match the complete combination count from three distinct primes.
• Only 8 is consistent with the prime factorisation 3 * 13 * 239.

Common Pitfalls:
Typical mistakes include stopping factorisation too early, miscalculating a quotient, or forgetting to check whether the last factor is prime. Another error is misusing the factor counting formula, for example adding exponents instead of adding one to each exponent and multiplying. Always write the prime factorisation clearly with exponents, then apply the formula carefully.

Final Answer:
The number 9321 has 8 positive factors.