`N = 4^(61) + 4^(61) + 4^(62) + 4^(62)`. Choose the correct statement(s):
I. N is divisible by 3.
II. N is divisible by 5.
III N is a perfect square.

Options

A. only I
B. only II
C. only III
D. only I and II

Answer

To solve the problem, we need to analyze the expression given for \( N \):

\[ N = 4^{61} + 4^{61} + 4^{62} + 4^{62} \]

### Step 1: Simplify the expression
First, we can combine like terms. Notice that there are two \( 4^{61} \) terms and two \( 4^{62} \) terms.

\[ N = 2 \cdot 4^{61} + 2 \cdot 4^{62} \]

### Step 2: Factor out the common term
Next, we can factor out \( 4^{61} \) from the expression:

\[ N = 4^{61} (2 + 2 \cdot 4) \]

### Step 3: Simplify the expression inside the parentheses
Now, simplify the expression inside the parentheses:

\[ N = 4^{61} (2 + 8) = 4^{61} \cdot 10 \]

### Step 4: Analyze divisibility by 3
Now we check if \( N \) is divisible by 3.

- \( 10 \) is not divisible by 3.
- \( 4^{61} \) is also not divisible by 3 (since \( 4 \mod 3 = 1 \)).

Thus, \( N \) is **not divisible by 3**.

### Step 5: Analyze divisibility by 5
Next, we check if \( N \) is divisible by 5.

- Since \( 10 \) is divisible by 5, and \( 4^{61} \) does not affect this divisibility, \( N \) is **divisible by 5**.

### Step 6: Check if \( N \) is a perfect square
Finally, we check if \( N \) is a perfect square.

- We can rewrite \( N \) as \( N = 10 \cdot (2^2)^{61} = 10 \cdot 2^{122} \).
- The term \( 2^{122} \) is a perfect square, but \( 10 \) is not a perfect square (since \( 10 = 2 \cdot 5 \), neither 2 nor 5 is a perfect square).

Thus, \( N \) is **not a perfect square**.

### Conclusion
Based on our analysis:
- I. \( N \) is not divisible by 3. (False)
- II. \( N \) is divisible by 5. (True)
- III. \( N \) is not a perfect square. (False)

The only correct statement is II.