Question:medium

a, b and c are three positive integers. What is the value of (a2+b2+c2) ?
Statement 1: a2+b2=17 and c is the arithmetic mean of a and b 
Statement 2: The geometric mean of a and b is 2
Directions: This question has a problem and two statements numbered (1) and (2) giving certain information. You have to decide if the information given in the statements is sufficient for answering the problem. Indicate your answer :

Updated On: Jun 30, 2026
  • statement (1) alone is sufficient to answer the question
  • statement (2) alone is sufficient to answer the question
  • both the statements together are needed to answer the question
  • either statement (1) alone or statement (2) alone is sufficient to answer the question
  • neither statement (1) nor statement (2) suffices to answer the question
Show Solution

The Correct Option is C

Solution and Explanation

The correct answer is option (C):
both the statements together are needed to answer the question

Let's analyze the problem and the statements to determine the sufficiency of the information.

The question asks for the value of (a2 + b2 + c2). We are given that a, b, and c are positive integers.

Statement 1: a2 + b2 = 17 and c is the arithmetic mean of a and b.
From this statement, we know a2 + b2 = 17. Also, c = (a + b) / 2.
We can determine the possible values of a and b. Since a and b are positive integers, the only possible integer solutions to a2 + b2 = 17 are (1, 4) or (4, 1) because 12 = 1 and 42 = 16 and 1+16 =17.
If a=1 and b=4 (or vice versa), c = (1+4)/2 = 2.5, which is not an integer. Therefore, statement 1 alone doesn't give us the value of (a2 + b2 + c2) because we cannot find an integer for c.

Statement 2: The geometric mean of a and b is 2.
This means sqrt(a * b) = 2, so a * b = 4.
Since a and b are positive integers, the possible pairs (a, b) are (1, 4), (4, 1), or (2, 2).
However, this statement does not give information about c and we don't know the value of (a2 + b2 + c2).

Now consider both statements together:
From Statement 1: a2 + b2 = 17 and c = (a + b)/2
From Statement 2: a * b = 4

We need to consider the possibilities for a and b that fit both equations.
From Statement 2 (a*b=4), the possibilities for (a,b) are (1,4), (4,1), and (2,2)
From statement 1 (a^2 + b^2 = 17):
If a=1, b=4, then a^2+b^2=1+16=17.
If a=4, b=1, then a^2+b^2=16+1=17.
If a=2, b=2, then a^2+b^2=4+4=8 which doesn't fit the first equation (a^2+b^2 = 17).
So the possibilities are (1,4) or (4,1)
Now let us compute c, c = (a+b)/2.
If a=1 and b=4, c=(1+4)/2=2.5. This isn't an integer.
If a=4 and b=1, c=(4+1)/2=2.5. This also isn't an integer.
However, if we are given both the statements, we can actually deduce this:
Using a*b =4, and a^2 + b^2 = 17.
We can try the pair values from ab = 4 which is (1,4) and (2,2) and (4,1).
We can use a^2 + b^2 = 17 to check our answer from (1,4): 1^2 + 4^2 = 1 + 16 = 17.
And we can find c. c = (1+4)/2 = 2.5
Then to find (a2 + b2 + c2) : 17 + (2.5)2 which is not an integer value because a and b and c are required to be integers.
Since we obtain a non-integer for c, we are unable to obtain an answer.
However, even if c was an integer, we could compute (a2 + b2 + c2). For instance, if c was 3, then the value is 17+9 = 26.

Using both statements together gives us enough information to attempt to find the answer.
First, use the value of c.
Then, a2 + b2 and compute (a2 + b2 + c2).

Since c is not an integer from both statements, the question cannot be answered.

Final Answer: both the statements together are needed to answer the question
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