Question

What is the value of prime number $x$?
I. $x^2 + x$ is a two-digit number greater than 50.
II. $x^3$ is a three-digit number.

• A. If the question can be answered with the help of statement I alone.
• B. If the question can be answered with the help of statement II alone.
• C. If both the statement I and statement II are needed to answer the question.
• D. If the question cannot be answered even with the help of both the statements.

Hint:

Use both numeric bounds and properties (like being prime) together to narrow down values step-by-step.

Answer 1

The Correct Option is C

From Statement I: We are told $x^2 + x$ is a two-digit number greater than 50. Let's try a few prime numbers: \[ \text{If } x = 7,\quad x^2 + x = 49 + 7 = 56 \quad \text{✓ valid}
\text{If } x = 11,\quad x^2 + x = 121 + 11 = 132 \quad \text{✗ three-digit} \] So this suggests $x$ is somewhere between 7 and 10. But that alone isn't enough. Now consider: From Statement II: $x^3$ is a three-digit number. That means: \[ x^3 \geq 100 \Rightarrow x \geq 5
x^3<1000 \Rightarrow x<10 \Rightarrow 5 \leq x<10 \] That gives us possible prime numbers in that range: 5 and 7. Now combine both statements: - $x=5$ gives $x^2+x=25+5=30$ too small - $x=7$ gives $x^2+x=49+7=56$ So $x=7$ is the only prime satisfying both conditions.