Which of the following is/are true?
(i) `43^3 - 1` is divisible by 11
(ii) `56^2 + 1` is divisible by19
(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.

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

1. We can rewrite this expression using the difference of cubes formula:
\[ a^3 - b^3 = (a - b)(a^2 + ab + b^2) \]
Here, \( a = 43 \) and \( b = 1 \).
So, we have:
\[ 43^3 - 1^3 = (43 - 1)(43^2 + 43 \cdot 1 + 1^2) = 42(43^2 + 43 + 1) \]

2. Now, we need to check if \( 42 \) is divisible by \( 11 \):
\[ 42 \div 11 = 3.8181 \quad (\text{not an integer}) \]
Therefore, \( 42 \) is not divisible by \( 11 \).

**Conclusion for Statement (i)**: False.

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

1. First, calculate \( 56^2 \):
\[ 56^2 = 3136 \]
Now, add \( 1 \):
\[ 3136 + 1 = 3137 \]

2. Now, check if \( 3137 \) is divisible by \( 19 \):
\[ 3137 \div 19 = 165.1 \quad (\text{not an integer}) \]
Therefore, \( 3137 \) is not divisible by \( 19 \).

**Conclusion for Statement (ii)**: False.

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

1. We can rewrite this expression using the difference of squares formula:
\[ a^2 - b^2 = (a - b)(a + b) \]
Here, \( a = 50 \) and \( b = 1 \):
\[ 50^2 - 1^2 = (50 - 1)(50 + 1) = 49 \cdot 51 \]

2. Now, check if \( 49 \cdot 51 \) is divisible by \( 17 \):
- \( 49 \div 17 = 2.882 \quad (\text{not an integer}) \)
- \( 51 \div 17 = 3 \quad (\text{is an integer}) \)

Since \( 51 \) is divisible by \( 17 \), \( 49 \cdot 51 \) is divisible by \( 17 \).

**Conclusion for Statement (iii)**: True.

### Statement (iv): `(729)^5 - 729` is divisible by 5.

1. We can factor this expression:
\[ 729^5 - 729 = 729(729^4 - 1) \]

2. Now, we need to check if \( 729^4 - 1 \) is divisible by \( 5 \):
- First, calculate \( 729 \mod 5 \):
\[ 729 \div 5 = 145.8 \quad (\text{remainder } 4) \]
So, \( 729 \equiv 4 \mod 5 \).

3. Now, calculate \( 729^4 \mod 5 \):
\[ 4^4 \mod 5 = 256 \mod 5 = 1 \]
Thus, \( 729^4 - 1 \equiv 1 - 1 \equiv 0 \mod 5 \).

**Conclusion for Statement (iv)**: True.

### Final Results:
- Statement (i): False
- Statement (ii): False
- Statement (iii): True
- Statement (iv): True