Question

Consider four natural numbers: $x$, $y$, $x+y$, and $x-y$. Two statements are given:
I. All four numbers are prime numbers.
II. The arithmetic mean of the numbers is greater than 4.
Which of the following statements would be sufficient to determine the sum of the four numbers?

• A. Statement I.
• B. Statement II.
• C. Statement I and Statement II.
• D. Neither Statement I nor Statement II.
• E. Either Statement I or Statement II.

Hint:

For number-based sufficiency problems, test small prime values systematically. Often uniqueness is forced by prime restrictions, while mean or inequality conditions rarely fix a unique sum.

Answer 1

The Correct Option is A

Step 1: Represent the numbers.
We have four numbers: $x$, $y$, $x+y$, and $x-y$. Their sum is \[ S = x + y + (x+y) + (x-y) = 3x + y. \] Step 2: Use Statement I.
All four numbers must be prime. That means:
- $x$ is prime.
- $y$ is prime.
- $x+y$ is prime.
- $x-y$ is prime.
Now $x-y$ is prime → $x > y$.
Step 3: Try small primes.
Let $y=2$, the smallest prime. Then $x-y=x-2$ must be prime. If $x=5$, then $x-y=3$ (prime), $x+y=7$ (prime). All four are prime: $x=5$, $y=2$, $x+y=7$, $x-y=3$.
Step 4: Compute the sum.
\[ S = 5 + 2 + 7 + 3 = 17. \] Thus, using Statement I alone, we uniquely determine the sum as 17.
Step 5: Check Statement II.
“The arithmetic mean > 4” only gives $S/4>4 \Rightarrow S>16$. Many possibilities remain; the sum is not uniquely determined. \[ \boxed{\text{Hence only Statement I is sufficient.}} \]