Question

Esther is giving Christmas presents to her family members. Each family member gets the same number of presents and no presents were leftover. If each family member gets at least one present, did each family member receive more than one present?
(1) Esther has forty Christmas presents to give out.
(2) If the number of family members were doubled, it would not be possible for each family member to get at least one present.

• 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 Yes/No questions, a definitive "No" is just as sufficient as a definitive "Yes". The key is whether the statement removes all ambiguity.

Answer 1

The Correct Option is B

Step 1: Understanding the Concept:
This is a Yes/No Data Sufficiency question. Let \(P\) be the total number of presents, \(F\) be the number of family members, and \(n\) be the number of presents each member receives. We are given \(P = F \times n\) and \(n \geq 1\). The question asks: Is \(n>1\)?
Step 2: Key Formula or Approach:
We need to analyze each statement to see if it provides a definitive "Yes" or "No" answer to the question "Is \(n>1\)?".
Step 3: Detailed Explanation:
Analyzing Statement (1):
"Esther has forty Christmas presents to give out."
This tells us \(P = 40\). So, \(40 = F \times n\). Both \(F\) and \(n\) must be integer factors of 40.
Can \(n>1\)? Yes. For example, if there are \(F=10\) family members, each gets \(n=4\) presents. Here, \(n>1\). The answer is "Yes".
Can \(n\) be equal to 1? Yes. If there are \(F=40\) family members, each gets \(n=1\) present. Here, \(n\) is not greater than 1. The answer is "No".
Since we can get both a "Yes" and a "No" answer, statement (1) is not sufficient.
Analyzing Statement (2):
"If the number of family members were doubled, it would not be possible for each family member to get at least one present."
The doubled number of family members is \(2F\). If it's not possible for each to get at least one present, it means the number of presents \(P\) is less than the doubled number of family members.
This gives us the inequality: \(P<2F\).
From the main problem, we know that \(P = F \times n\). We can substitute this into the inequality:
\[ F \times n<2F \] Since \(F\) (the number of family members) must be a positive integer, we can safely divide both sides by \(F\) without changing the inequality direction:
\[ n<2 \] The problem also states that each family member gets at least one present, which means \(n \geq 1\).
Combining these two conditions, we have \(1 \leq n<2\). Since \(n\) must be an integer (number of presents), the only possible value for \(n\) is 1.
If we know for certain that \(n=1\), we can definitively answer the question "Is \(n>1\)?". The answer is a definite "No".
Therefore, statement (2) is sufficient.
Step 4: Final Answer:
Statement (2) alone is sufficient, but statement (1) alone is not.