Question

x is a positive integer less than 20. What is the value of x?
1. x is the sum of two consecutive integers.
2. x is the sum of five consecutive integers.

• A. Statement (2) ALONE is sufficient, but statement (1) alone is not sufficient to answer the question asked
• B. EACH statement ALONE is sufficient to answer the question asked
• C. Statements (1) and (2) TOGETHER are NOT sufficient to answer the question asked, and additional data specific to the problem are needed
• D. Both statements (1) and (2) TOGETHER are sufficient to answer the question asked; but NEITHER statement ALONE is sufficient
• E. Statement (1) ALONE is sufficient, but statement (2) alone is not sufficient to answer the question asked

Hint:

For problems involving sums of consecutive integers, it's useful to know these properties: - The sum of 'k' consecutive integers is a multiple of 'k' if 'k' is odd. - The sum of 'k' consecutive integers is never a multiple of 'k' if 'k' is even (it's a multiple of k/2, but not an integer multiple of k). - The sum of two consecutive integers is always odd.

Answer 1

The Correct Option is C

Step 1: Understanding the Concept:
The question asks for the specific value of x, which is a positive integer less than 20. We need to use the properties described in the statements to narrow down the possibilities for x. A statement (or combination) is sufficient only if it leads to a single, unique value for x.

Step 2: Key Formula or Approach:
We will translate each statement into an algebraic property of x and then list all possible values of x that satisfy the given conditions (\(0 < x < 20\)).

Step 3: Detailed Explanation:
From the question stem, we know that \(x \in \{1, 2, 3, \dots, 19\}\).
Analyzing Statement (1): x is the sum of two consecutive integers.
Let the two consecutive integers be \(n\) and \(n+1\).
\[ x = n + (n+1) = 2n + 1 \] This means that x must be an odd number.
The possible values for x (positive odd integers less than 20) are: \(\{1, 3, 5, 7, 9, 11, 13, 15, 17, 19\}\).
Since there is more than one possible value for x, statement (1) alone is not sufficient.
Analyzing Statement (2): x is the sum of five consecutive integers.
Let the five consecutive integers be \(n, n+1, n+2, n+3, n+4\).
\[ x = n + (n+1) + (n+2) + (n+3) + (n+4) = 5n + 10 = 5(n+2) \] This means that x must be a multiple of 5.
Since x must be a positive integer, \(5(n+2) > 0\), which means \(n+2 > 0\), so \(n > -2\). The smallest possible value for n is -1.
Let's find the possible values for x:
If \(n=-1\), \(x = 5(-1+2) = 5\). The integers are -1, 0, 1, 2, 3.
If \(n=0\), \(x = 5(0+2) = 10\). The integers are 0, 1, 2, 3, 4.
If \(n=1\), \(x = 5(1+2) = 15\). The integers are 1, 2, 3, 4, 5.
If \(n=2\), \(x = 5(2+2) = 20\), which is not less than 20.
So, the possible values for x are: \(\{5, 10, 15\}\).
Since there is more than one possible value for x, statement (2) alone is not sufficient.
Analyzing Both Statements Together:
From statement (1), x must be odd.
From statement (2), x must be a multiple of 5.
We need to find the numbers that are in both sets of possibilities.
Possible values from (1): \(\{1, 3, 5, 7, 9, 11, 13, 15, 17, 19\}\)
Possible values from (2): \(\{5, 10, 15\}\)
The common values are \(\{5, 15\}\).
Even with both statements, there are still two possible values for x (5 and 15). We cannot determine a unique value for x.
Therefore, both statements together are not sufficient.

Step 4: Final Answer:
Statements (1) and (2) together are not sufficient to answer the question.