Consider the nine digit number n = 7 3 `alpha` 4 9 6 1 `beta` 0. The number of possible values of `beta` for wich `i^(N) = 1 ("where " i=sqrt-1)`,

Options

A. 2
B. 3
C. 4
D. 5

Answer

To solve the problem, we need to determine the possible values of the digit `β` in the nine-digit number \( n = 73\alpha4961\beta0 \) such that \( i^n = 1 \), where \( i = \sqrt{-1} \).

### Step-by-Step Solution:

1. **Understanding the Condition**:
The condition \( i^n = 1 \) implies that \( n \) must be a multiple of 4. This is because \( i^4 = 1 \) and the powers of \( i \) repeat every four numbers: \( i, -1, -i, 1 \).

2. **Identifying \( n \)**:
The number \( n \) can be expressed as \( 73\alpha4961\beta0 \). The last two digits of \( n \) are \( \beta0 \).

3. **Divisibility by 4**:
For \( n \) to be a multiple of 4, the last two digits \( \beta0 \) must be divisible by 4. This means we need to check the values of \( \beta0 \) for \( \beta = 0, 1, 2, ..., 9 \).

4. **Calculating Possible Values**:
The last two digits \( \beta0 \) represent the numbers:
- \( 00 \) (when \( \beta = 0 \))
- \( 10 \) (when \( \beta = 1 \))
- \( 20 \) (when \( \beta = 2 \))
- \( 30 \) (when \( \beta = 3 \))
- \( 40 \) (when \( \beta = 4 \))
- \( 50 \) (when \( \beta = 5 \))
- \( 60 \) (when \( \beta = 6 \))
- \( 70 \) (when \( \beta = 7 \))
- \( 80 \) (when \( \beta = 8 \))
- \( 90 \) (when \( \beta = 9 \))

5. **Checking Each Value for Divisibility by 4**:
- \( 00 \div 4 = 0 \) (divisible)
- \( 10 \div 4 = 2.5 \) (not divisible)
- \( 20 \div 4 = 5 \) (divisible)
- \( 30 \div 4 = 7.5 \) (not divisible)
- \( 40 \div 4 = 10 \) (divisible)
- \( 50 \div 4 = 12.5 \) (not divisible)
- \( 60 \div 4 = 15 \) (divisible)
- \( 70 \div 4 = 17.5 \) (not divisible)
- \( 80 \div 4 = 20 \) (divisible)
- \( 90 \div 4 = 22.5 \) (not divisible)

6. **Listing the Valid Values of \( \beta \)**:
The valid values of \( \beta \) that make \( \beta0 \) divisible by 4 are:
- \( \beta = 0 \) (for \( 00 \))
- \( \beta = 2 \) (for \( 20 \))
- \( \beta = 4 \) (for \( 40 \))
- \( \beta = 6 \) (for \( 60 \))
- \( \beta = 8 \) (for \( 80 \))

7. **Counting the Possible Values**:
Thus, the possible values of \( \beta \) are \( 0, 2, 4, 6, 8 \), giving us a total of 5 possible values.

### Final Answer:
The number of possible values of \( \beta \) for which \( i^n = 1 \) is **5**.