Question

What is the value of positive two-digit integer x?
(1) The sum of the two digits is 5.
(2) x is prime.

• A. Statement (1) ALONE is sufficient, but statement (2) alone is not sufficient.
• B. Statement (2) ALONE is sufficient, but statement (1) alone is not sufficient.
• C. BOTH statements TOGETHER are sufficient, but NEITHER statement ALONE is sufficient.
• D. EACH statement ALONE is sufficient.
• E. Statements (1) and (2) TOGETHER are NOT sufficient.

Hint:

For "what is the value" questions, your goal is to find one and only one possible answer. If you are left with two or more possibilities after using all the information, the data is insufficient.

Answer 1

Answer: (E)

Step 1: Understanding the Concept:
This is a Data Sufficiency problem asking for the specific value of a two-digit integer \(x\). A statement or combination of statements is sufficient only if it narrows down the possibilities to a single value for \(x\).
Step 2: Key Formula or Approach:
We will list the possible values of \(x\) that satisfy each statement and then find the intersection of those lists when combining the statements.
Step 3: Detailed Explanation:
Analyzing Statement (1):
"The sum of the two digits is 5."
Let the two-digit integer be represented as \(10a + b\), where \(a\) is the tens digit and \(b\) is the units digit. We are given \(a+b=5\). Since \(x\) is a positive two-digit integer, \(a\) cannot be 0.
Possible pairs of digits (a, b) are (1, 4), (2, 3), (3, 2), (4, 1), (5, 0).
This gives the following possible values for \(x\): 14, 23, 32, 41, 50.
Since there are multiple possible values for \(x\), statement (1) is not sufficient.
Analyzing Statement (2):
"x is prime."
A prime number is a number greater than 1 that has only two divisors: 1 and itself. There are many two-digit prime numbers, such as 11, 13, 17, 19, 23, etc.
Since there are many possibilities for \(x\), statement (2) is not sufficient.
Analyzing Both Statements Together:
We need to find the numbers from the list in statement (1) that are also prime.
The list is: {14, 23, 32, 41, 50}.
Let's check each for primality:
- 14 is not prime (divisible by 2).
- 23 is a prime number.
- 32 is not prime (divisible by 2).
- 41 is a prime number.
- 50 is not prime (divisible by 5).
After combining both statements, the possible values for \(x\) are 23 and 41.
Since there are still two possible values, we cannot determine a unique value for \(x\).
Therefore, even together, the statements are not sufficient.
Step 4: Final Answer:
Statements (1) and (2) together are not sufficient to determine the value of x.