Question

If a, b, and c are distinct positive integers where a$<$b$<$c and $\sqrt{abc}=c$, what is the value of a?
1. c=8
2. The average of a, b, and c is 143

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

Hint:

In Data Sufficiency problems involving number properties, always start by simplifying the initial conditions given in the question stem. Here, simplifying $\sqrt{abc}=c$ to $ab=c$ and then deriving the constraint $a>1$ is crucial for evaluating the statements efficiently.

Answer 1

The Correct Option is C

Step 1: Understanding the Concept:
This is a data sufficiency question involving number properties. We are given a relationship between three distinct positive integers \(a, b,\) and \(c\) and are asked to find the value of \(a\). We need to analyze each statement to see if it provides enough information to find a unique value for \(a\).

Step 2: Key Formula or Approach:
The given equation is \(\sqrt{abc} = c\).
Since \(c\) is a positive integer, we can square both sides:
\[ (\sqrt{abc})^2 = c^2 \] \[ abc = c^2 \] Since \(c > 0\), we can divide both sides by \(c\):
\[ ab = c \] We are also given that \(a, b, c\) are distinct positive integers with the condition \(a < b < c\).
Substituting \(c = ab\) into the inequality gives \(a < b < ab\).
Since \(a\) is a positive integer, \(a \geq 1\).
The inequality \(b < ab\) simplifies to \(1 < a\) (by dividing by \(b\), which is positive). So, \(a\) must be greater than 1, i.e., \(a \ge 2\). If \(a=1\), \(1<b<b\), which is impossible. So \(a \neq 1\). Wait, let's recheck. \(a,b,c\) are distinct positive integers. \(ab=c\). We also need \(a<b<c\). Substituting \(c=ab\) into \(b<c\) gives \(b < ab\). Since \(b\) is positive, we can divide by \(b\) to get \(1 < a\). So \(a\) must be at least 2. This is a key constraint derived from the problem statement itself.

Step 3: Detailed Explanation:
Analyzing Statement (1): c = 8
Using the derived relationship \(ab = c\), we have:
\[ ab = 8 \] Since \(a\) and \(b\) are distinct positive integers and \(a < b\), we need to find pairs of factors of 8.
The possible pairs \((a, b)\) are \((1, 8)\) and \((2, 4)\).
Now we check these pairs against the condition \(a < b < c\). Here, \(c=8\).
Case 1: \((a, b) = (1, 8)\). The condition becomes \(1 < 8 < 8\). This is false because \(b\) is not less than \(c\).
Case 2: \((a, b) = (2, 4)\). The condition becomes \(2 < 4 < 8\). This is true.
In this case, the value of \(a\) is uniquely determined as 2.
Therefore, Statement (1) ALONE is sufficient.
Analyzing Statement (2): The average of a, b, and c is 143
The average is given by \(\frac{a+b+c}{3} = 143\).
This means \(a+b+c = 3 \times 143 = 429\).
We also know \(c = ab\). Substituting this into the sum equation:
\[ a+b+ab = 429 \] To solve this, we can use a factoring trick:
\[ ab + a + b + 1 = 429 + 1 \] \[ (a+1)(b+1) = 430 \] We need to find integer factors of 430, where \(a+1\) and \(b+1\) are the factors. Since \(a < b\), we must have \(a+1 < b+1\).
The prime factorization of 430 is \(2 \times 5 \times 43\). The factors are 1, 2, 5, 10, 43, 86, 215, 430.
We know from the initial analysis that \(a > 1\), so \(a+1 > 2\).
Let's test the factor pairs of 430 for \((a+1, b+1)\):
Case 1: \(a+1 = 5\). Then \(a=4\). And \(b+1 = 86\), so \(b=85\). Let's find \(c\): \(c = ab = 4 \times 85 = 340\). Check the condition \(a < b < c\): \(4 < 85 < 340\). This is a valid solution. Here, \(a=4\).
Case 2: \(a+1 = 10\). Then \(a=9\). And \(b+1 = 43\), so \(b=42\). Let's find \(c\): \(c = ab = 9 \times 42 = 378\). Check the condition \(a < b < c\): \(9 < 42 < 378\). This is also a valid solution. Here, \(a=9\).
Since we found at least two possible values for \(a\) (4 and 9), this statement does not give a unique value for \(a\).
Therefore, Statement (2) ALONE is not sufficient.

Step 4: Final Answer:
Statement (1) alone is sufficient to determine a unique value for \(a\), but statement (2) alone is not. Therefore, the correct option is (C).