Which of the following is/are true?
(i) `43^(3)-1` is divisible by 11
(ii) `56^(2)+1` is divisible by 19
(iii) `50^(2)-1` is divisible by 17
(iv)`(729)^(5)-729` is divisible by 5

Options

A. (i) and (ii)
B. (iii) and (iv)
C. (ii), (iii) and (iv)
D. (ii) and (iii)

Answer

To determine which of the statements are true, we will evaluate each statement one by one.

### Step 1: Evaluate Statement (i) `43^(3) - 1` is divisible by 11

We can use Fermat's Little Theorem, which states that if \( p \) is a prime number and \( a \) is an integer not divisible by \( p \), then \( a^{p-1} \equiv 1 \mod p \).

- Here, \( a = 43 \) and \( p = 11 \).
- Since \( 43 \mod 11 = 10 \), we can rewrite \( 43^3 - 1 \) as \( 10^3 - 1 \).

Calculating \( 10^3 - 1 \):
\[ 10^3 = 1000 \implies 1000 - 1 = 999 \]

Now, we check if \( 999 \) is divisible by \( 11 \):
\[ 999 \div 11 = 90.8181 \quad (\text{not an integer}) \]
Thus, \( 999 \) is not divisible by \( 11 \).

**Conclusion**: Statement (i) is **false**.

### Step 2: Evaluate Statement (ii) `56^(2) + 1` is divisible by 19

We will evaluate \( 56^2 + 1 \) modulo \( 19 \).

Calculating \( 56 \mod 19 \):
\[ 56 \div 19 = 2.947 \implies 56 - 2 \times 19 = 56 - 38 = 18 \]
So, \( 56 \equiv 18 \mod 19 \).

Now, calculate \( 56^2 + 1 \):
\[ 56^2 \equiv 18^2 \mod 19 \]
Calculating \( 18^2 \):
\[ 18^2 = 324 \]
Now, we find \( 324 \mod 19 \):
\[ 324 \div 19 = 17.0526 \quad \Rightarrow 324 - 17 \times 19 = 324 - 323 = 1 \]
Thus, \( 18^2 \equiv 1 \mod 19 \).

Now, adding \( 1 \):
\[ 56^2 + 1 \equiv 1 + 1 = 2 \mod 19 \]
Since \( 2 \) is not divisible by \( 19 \), we conclude that:

**Conclusion**: Statement (ii) is **false**.

### Step 3: Evaluate Statement (iii) `50^(2) - 1` is divisible by 17

We can use the difference of squares:
\[ 50^2 - 1 = (50 - 1)(50 + 1) = 49 \times 51 \]

Now, we check if \( 49 \times 51 \) is divisible by \( 17 \):
- \( 49 \mod 17 = 15 \)
- \( 51 \mod 17 = 0 \)

Since \( 51 \) is divisible by \( 17 \), it follows that \( 49 \times 51 \) is divisible by \( 17 \).

**Conclusion**: Statement (iii) is **true**.

### Step 4: Evaluate Statement (iv) `(729)^(5) - 729` is divisible by 5

We can factor this expression:
\[ 729^5 - 729 = 729(729^4 - 1) \]
Now, we need to check if \( 729 \) is divisible by \( 5 \):
\[ 729 \mod 5 = 4 \]
Thus, \( 729 \) is not divisible by \( 5 \).

Next, we check \( 729^4 - 1 \):
Using Fermat's Little Theorem again:
\[ 729 \equiv 4 \mod 5 \]
Calculating \( 4^4 \mod 5 \):
\[ 4^4 = 256 \implies 256 \mod 5 = 1 \]
Thus, \( 729^4 \equiv 1 \mod 5 \), and:
\[ 729^4 - 1 \equiv 1 - 1 = 0 \mod 5 \]
Thus, \( 729^4 - 1 \) is divisible by \( 5 \).

Since \( 729^5 - 729 = 729(729^4 - 1) \) and \( 729^4 - 1 \) is divisible by \( 5 \), we conclude that:

**Conclusion**: Statement (iv) is **true**.

### Final Summary of Statements:
- Statement (i): False
- Statement (ii): False
- Statement (iii): True
- Statement (iv): True

### Final Answer:
The true statements are (iii) and (iv), which corresponds to option 2.