Consider the composite number X = p^3 \cdot q^2, where p and q are distinct prime numbers.… — State PSC Exam

Consider the composite number $X = p^3 \cdot q^2$ , where $p$ and $q$ are distinct prime numbers. Which of the following statements regarding the factors and properties of $X$ are correct?

The total number of positive factors of $X$ is exactly $12$ .Answer
If $p=2$ and $q=3$ , the sum of all positive factors of $X$ is $195$ .Answer
C
The total number of positive factors can be derived by the expression $3 + 1 \times 2 + 1$ , which yields exactly $6$ factors.
D
The total number of positive factors of $X$ that are multiples of either $p$ or $q$ is $17$ , calculated by directly adding the $9$ factors divisible by $p$ to the $8$ factors divisible by $q$ .

Answer

The correct statements are that the total number of positive factors is exactly 12, and if p=2 and q=3, the sum of all positive factors is 195.

The correct options properly apply the theorems of prime factorization. A composite number with prime factorization

p^a \cdot q^b

always has

(a+1)(b+1)

total factors, which confirms the count of

12

. Furthermore, the geometric series expansion for the sum of factors correctly evaluates to

195

when substituting

p=2

and

q=3

Step-by-Step Solution

Determine the total number of positive factors of X.

The prime powers are 3 and 2. Using the formula (a+1)(b+1), we get (3+1)(2+1) = 4 * 3 = 12.

This establishes the baseline count of factors necessary for evaluating the first statement and the arithmetic logic of the third statement.

Calculate the sum of factors for the specific case where p=2 and q=3.

The sum is (1 + 2 + 4 + 8) * (1 + 3 + 9) = 15 * 13 = 195.

This verifies the explicit calculation claim made in the second statement.

Analyze the arithmetic expression proposed for factor counting: 3 + 1 * 2 + 1.

The expression lacks brackets. By standard order of operations, 3 + (1*2) + 1 = 6, which is mathematically flawed for factor counting.

Identifying the missing parentheses reveals a fundamental BODMAS order of operations violation.

Apply the inclusion-exclusion principle to count factors that are multiples of p or q.

Factors divisible by p: 9. Factors divisible by q: 8. Intersection (divisible by pq): 6. Union = 9 + 8 - 6 = 11.

This verifies whether directly adding the two sets is mathematically sound, proving that the fourth statement double-counts the overlap.

Key Concept

Prime factorization properties, including total factor counting, sum of factors, and applying set theory (inclusion-exclusion) to factor subsets.

Estimated Time:1m 30s

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