Let `U` be set with number of elements in `U` is `2009`.
Consider the following statements :
`I` : If `A`, `B` are subsets of `U` with `n(AuuB)=280`, then `n(A'nnB')=x_(1)^(3)+x_(2)^(3)=y_(1)^(3)+y_(2)^(3)`
for some positive integers `x_(1)`, `x_(2)y_(1)`, `y_(2)`
`II` : If `A` is a subset of `U`, with `n(A)=1681` and out of these `1681` elements, exactly `1075` elements belong to a subset `B` of `U`, then `n(A-B)=m^(2)+p_(1)p_(2)p_(3)` for some positive integer `m` and distinct primes `p_(1)`, `p_(2)`, `p_(3)`
Which of the statements given above is/are correct ?

Options

A. `I` only
B. `II` only
C. Both `I` and `II`
D. Neither `I` nor `II`

Answer

To solve the given problem, we will analyze both statements step by step.

### Step 1: Analyze Statement I
We are given:
- The universal set \( U \) has \( n(U) = 2009 \).
- The number of elements in the union of sets \( A \) and \( B \) is \( n(A \cup B) = 280 \).

We need to find \( n(A' \cap B') \), which is the number of elements in the complement of the union of \( A \) and \( B \):
\[ n(A' \cap B') = n(U) - n(A \cup B) \]
Substituting the values:
\[ n(A' \cap B') = 2009 - 280 = 1729 \]

Next, we need to express \( 1729 \) as a sum of cubes of two positive integers:
\[ 1729 = x_1^3 + x_2^3 \]
We can find that:
\[ 1^3 + 12^3 = 1 + 1728 = 1729 \]
Thus, one possible solution is \( x_1 = 1 \) and \( x_2 = 12 \).

Now, we also need to express \( n(A \cap B) \) as a sum of cubes:
Using the formula for the union of two sets:
\[ n(A \cup B) = n(A) + n(B) - n(A \cap B) \]
However, we don't have \( n(A) \) and \( n(B) \) directly, but we can assume \( n(A \cap B) = y_1^3 + y_2^3 \) for some integers \( y_1 \) and \( y_2 \).

By checking known cube sums, we can find:
\[ 10^3 + 9^3 = 1000 + 729 = 1729 \]
Thus, \( y_1 = 10 \) and \( y_2 = 9 \).

### Conclusion for Statement I
Since both conditions can be satisfied, Statement I is **correct**.

### Step 2: Analyze Statement II
We are given:
- \( n(A) = 1681 \)
- \( n(A \cap B) = 1075 \)

We need to find \( n(A - B) \):
\[ n(A - B) = n(A) - n(A \cap B) \]
Substituting the values:
\[ n(A - B) = 1681 - 1075 = 606 \]

Now, we need to express \( 606 \) in the form:
\[ 606 = m^2 + p_1 p_2 p_3 \]
where \( m \) is a positive integer and \( p_1, p_2, p_3 \) are distinct primes.

We can check:
\[ 24^2 + 2 \times 3 \times 5 = 576 + 30 = 606 \]
Here, \( m = 24 \) and the distinct primes are \( 2, 3, 5 \).

### Conclusion for Statement II
Since we can express \( 606 \) in the required form, Statement II is also **correct**.

### Final Conclusion
Both statements I and II are correct.