Let `N = (10^143 -1)/9` Then which of the following is true?

Options

A. N is prime
B. ` N = (1+10^(12) + 10^(24) + …+10^(132))(1+10 + 10^(2) +…+10^(10))`
C. ` N = (1+10^(11) + 10^(22) + …+10^(132))(1+10 + 10^(2) +…+10^(10))`
D. ` N = (1+10^(13) + 10^(26) + …+10^(130))(1+10 + 10^(2) +…+10^(12))`

Answer

To solve the problem, we start with the expression given for \( N \):

\[ N = \frac{10^{143} - 1}{9} \]

### Step 1: Understanding the Expression
The expression \( 10^{143} - 1 \) can be interpreted as a number that consists of 143 nines. This is because \( 10^{143} \) is a 1 followed by 143 zeros, and subtracting 1 from it gives us 143 nines.

### Step 2: Simplifying \( N \)
Now, we can rewrite \( N \):

\[ N = \frac{999\ldots9}{9} \]

where there are 143 nines in the numerator. Dividing each 9 by 9 gives us:

\[ N = 111\ldots1 \]

where there are 143 ones in the result.

### Step 3: Analyzing the Value of \( N \)
The number \( N \) can be expressed as:

\[ N = 111\ldots1 \quad (\text{143 times}) \]

This number can also be represented in terms of powers of 10:

\[ N = 1 + 10 + 10^2 + \ldots + 10^{142} \]

This is a geometric series with the first term \( a = 1 \) and the common ratio \( r = 10 \). The formula for the sum of a geometric series is:

\[ S_n = \frac{a(r^n - 1)}{r - 1} \]

where \( n \) is the number of terms. Here, \( n = 143 \):

\[ N = \frac{1(10^{143} - 1)}{10 - 1} = \frac{10^{143} - 1}{9} \]

This confirms our earlier expression for \( N \).

### Step 4: Checking the Options
1. **Option 1: Is \( N \) prime?**
- \( N \) is a repunit (a number consisting solely of the digit 1). A repunit \( R_k \) is prime only if \( k \) is prime. Since \( 143 = 11 \times 13 \), \( N \) is not prime.

2. **Option 2: \( N = 1 + 10^{12} + 10^{24} + \ldots + 10^{32} \)**
- This expression does not correctly represent \( N \) because it does not yield the correct number of ones.

3. **Option 3: \( N = 1 + 10^{10} + 10^{20} + \ldots + 10^{132} \)**
- This expression also does not yield the correct number of ones.

4. **Option 4: \( N = 1 + 10^{11} + 10^{22} + \ldots + 10^{133} \)**
- This expression does not yield the correct number of ones either.

### Conclusion
From the analysis, we find that:
- **Option 1 is false** (N is not prime).
- **Options 2, 3, and 4 are also false** as they do not correctly represent \( N \).

Thus, the correct conclusion is that none of the options provided are true.