Question

Consider the following statements about set:
A. $35 \in \{x : x \text{ has exactly four positive factors}\}$}
B. $128 \in \{y : \text{the sum of all positive factors of } y \text{ is } 4y\}$}
C. $3 \notin \{x : x^4 - 5x^3 + 2x^2 - 112x + 6 = 0\}$}
D. $28 \in \{y : \text{the sum of all positive factors of } y \text{ is } 2y\}$} Choose the correct answer from the options given below:

• A. A, B, C only
• B. A, B only
• C. A, C, D only
• D. A, B, C, D

Hint:

A number is "Perfect" if the sum of its divisors equals $2n$. For $n=2^p-1(2^p-1)$, if $2^p-1$ is prime, then $n$ is perfect. For $p=3$, $n=4(7)=28$.

Answer 1

The Correct Option is C

Concept: This problem requires verifying membership in sets defined by specific mathematical properties, including factor counts, factor sums (perfect numbers), and polynomial roots.

Step 1: Evaluate Statement A.
The factors of 35 are 1, 5, 7, and 35. Total count = 4. Since 35 has exactly four positive factors, the statement is True.

Step 2: Evaluate Statement B.
128 is $2^7$. The factors are 1, 2, 4, 8, 16, 32, 64, 128. Sum = $1 + 2 + 4 + 8 + 16 + 32 + 64 + 128 = 255$. $4y = 4(128) = 512$. Since $255 \neq 512$, the statement is False.

Step 3: Evaluate Statement C.
Let $f(x) = x^4 - 5x^3 + 2x^2 - 112x + 6$. $f(3) = 3^4 - 5(3^3) + 2(3^2) - 112(3) + 6$ $f(3) = 81 - 5(27) + 2(9) - 336 + 6$ $f(3) = 81 - 135 + 18 - 336 + 6 = -366$. Since $f(3) \neq 0$, 3 is not an element of the set. The statement $3 \notin \{...\}$ is True.

Step 4: Evaluate Statement D.
Factors of 28: 1, 2, 4, 7, 14, 28. Sum = $1 + 2 + 4 + 7 + 14 + 28 = 56$. $2y = 2(28) = 56$. Since Sum $= 2y$, 28 is a perfect number and the statement is True. Conclusion: A, C, and D are correct.