Question:hard

a, b and c are three positive integers. What is the value of \(a^2+b^2+c^2\)?

Statement 1: \(a^2+b^2=17\) and c is the arithmetic mean of a and b
Statement 2: The geometric mean of a and b is 2

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Combine \(a^2+b^2\) with \(ab\) to get \((a+b)^2\), then use c as the arithmetic mean.
Updated On: Jul 21, 2026
  • If the data in statement (1) alone is sufficient to answer the question
  • If the data in statement (2) alone is sufficient to answer the question
  • If the data in both the statements together are needed to answer the question
  • If either statement (1) alone or statement (2) alone is sufficient to answer the question
Show Solution

The Correct Option is C

Solution and Explanation

Step 1: List what must be found.
The target is $a^2+b^2+c^2$, and c only has meaning once we know it is defined as the average of a and b from statement 1.

Step 2: Check statement 1 in isolation.
Statement 1 fixes $a^2+b^2 = 17$ and tells us $c$ equals the average of a and b.
A single equation in two unknowns, a and b, has many solutions, so $a+b$ is not fixed by this equation alone, which means $c$ is not fixed either.
Statement 1 by itself cannot produce one number for the answer.

Step 3: Check statement 2 in isolation.
Statement 2 only fixes the product $ab = 4$ through the geometric mean condition $\sqrt{ab}=2$.
It says nothing about $a^2+b^2$, and it gives no rule at all for c, since c's definition lives only in statement 1.
Statement 2 by itself is clearly not enough.

Step 4: Merge the two equations.
List the two equations together: $a^2+b^2=17$ and $ab=4$.
Subtracting twice the product from the sum of squares gives $(a-b)^2 = a^2+b^2-2ab = 17-8=9$, so $a-b=3$ (taking a as the larger positive integer).
Adding twice the product instead gives $(a+b)^2=17+8=25$, so $a+b=5$.
Solving the pair $a-b=3$ and $a+b=5$ gives $a=4$ and $b=1$, matching positive integers.
The mean $c=(a+b)/2 = 5/2$, so $a^2+b^2+c^2 = 17+25/4 = 93/4$.

Final Answer:
Only the combination of both statements pins down one numeric answer. \[ \boxed{\text{Option (c): Both statements together are needed}} \]
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