Consider the nine digit number n = 7 3 `alpha` 4 9 6 1 `beta` 0. If q is the number of all possible values of `beta` for which the given number is divisible by 8, then q is equal to

Options

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

Answer

To solve the problem, we need to determine the values of `beta` in the nine-digit number \( n = 7 3 \alpha 4 9 6 1 \beta 0 \) such that the number is divisible by 8.

### Step-by-Step Solution:

1. **Understanding Divisibility by 8**:
A number is divisible by 8 if the number formed by its last three digits is divisible by 8. In this case, the last three digits of our number are \( 1 \beta 0 \).

2. **Forming the Last Three Digits**:
The last three digits can be expressed as \( 1\beta0 \), which can be interpreted as \( 100 + 10\beta + 0 \) or simply \( 10\beta + 10 \).

3. **Finding Possible Values of `beta`**:
We need to find values of \( \beta \) such that \( 1\beta0 \) is divisible by 8. The possible values of \( \beta \) can be from 0 to 9 (i.e., \( \beta = 0, 1, 2, 3, 4, 5, 6, 7, 8, 9 \)).

4. **Calculating Last Three Digits for Each `beta`**:
We will check each value of \( \beta \):
- For \( \beta = 0 \): \( 100 \div 8 = 12.5 \) (not divisible)
- For \( \beta = 1 \): \( 110 \div 8 = 13.75 \) (not divisible)
- For \( \beta = 2 \): \( 120 \div 8 = 15 \) (divisible)
- For \( \beta = 3 \): \( 130 \div 8 = 16.25 \) (not divisible)
- For \( \beta = 4 \): \( 140 \div 8 = 17.5 \) (not divisible)
- For \( \beta = 5 \): \( 150 \div 8 = 18.75 \) (not divisible)
- For \( \beta = 6 \): \( 160 \div 8 = 20 \) (divisible)
- For \( \beta = 7 \): \( 170 \div 8 = 21.25 \) (not divisible)
- For \( \beta = 8 \): \( 180 \div 8 = 22.5 \) (not divisible)
- For \( \beta = 9 \): \( 190 \div 8 = 23.75 \) (not divisible)

5. **Identifying Valid Values of `beta`**:
From our calculations, the values of \( \beta \) that make \( 1\beta0 \) divisible by 8 are:
- \( \beta = 2 \)
- \( \beta = 6 \)

6. **Counting the Valid Values**:
Therefore, the total number of valid values for \( \beta \) is \( 2 \).

### Final Answer:
Thus, the value of \( q \) (the number of all possible values of \( \beta \)) is \( 2 \).