A number 774958 `N_(1)` 96 `N_(2)` to be divisible by 8 and 9, the values of `N_(1) and N_(2)` will be

Options

A. `7,8`
B. `0,8`
C. `5,8`
D. `6,7`

Answer

To solve the problem of finding the values of \( N_1 \) and \( N_2 \) in the number 774958N1N296 such that it is divisible by both 8 and 9, we will follow these steps:

### Step 1: Check divisibility by 8
A number is divisible by 8 if its last three digits form a number that is divisible by 8. The last three digits of our number are \( N_1N296 \).

To find \( N_1 \), we will evaluate the possible values of \( N_1 \) (0-9) to see which makes \( N_1N296 \) divisible by 8.

- The last three digits can be expressed as \( 100N_1 + 296 \).
- We will check \( 296, 396, 496, 596, 696, 796, 896, 996 \) (where \( N_1 \) takes values from 0 to 9).

Now we check each:

- **296 ÷ 8 = 37.0** (divisible)
- **396 ÷ 8 = 49.5** (not divisible)
- **496 ÷ 8 = 62.0** (divisible)
- **596 ÷ 8 = 74.5** (not divisible)
- **696 ÷ 8 = 87.0** (divisible)
- **796 ÷ 8 = 99.5** (not divisible)
- **896 ÷ 8 = 112.0** (divisible)
- **996 ÷ 8 = 124.5** (not divisible)

Thus, the possible values for \( N_1 \) that make the last three digits divisible by 8 are 0, 4, 6, and 8.

### Step 2: Check divisibility by 9
A number is divisible by 9 if the sum of its digits is divisible by 9. The sum of the digits in our number is:

\[ 7 + 7 + 4 + 9 + 5 + 8 + N_1 + N_2 + 2 + 9 + 6 = 57 + N_1 + N_2 \]

We need \( 57 + N_1 + N_2 \) to be divisible by 9.

Calculating \( 57 \mod 9 \):

\[ 57 \div 9 = 6 \quad \text{remainder} = 3 \]

Thus, \( 57 \equiv 3 \mod 9 \). We need:

\[ 3 + N_1 + N_2 \equiv 0 \mod 9 \]

This simplifies to:

\[ N_1 + N_2 \equiv 6 \mod 9 \]

### Step 3: Combine conditions
Now we combine the conditions from Steps 1 and 2.

Possible values for \( N_1 \) are 0, 4, 6, and 8. We will check each value to find corresponding \( N_2 \):

1. **If \( N_1 = 0 \)**:
- \( N_2 \equiv 6 \mod 9 \) → Possible \( N_2 = 6 \)

2. **If \( N_1 = 4 \)**:
- \( N_2 \equiv 2 \mod 9 \) → Possible \( N_2 = 2 \)

3. **If \( N_1 = 6 \)**:
- \( N_2 \equiv 0 \mod 9 \) → Possible \( N_2 = 0 \)

4. **If \( N_1 = 8 \)**:
- \( N_2 \equiv 6 \mod 9 \) → Possible \( N_2 = 6 \)

### Final Values
The pairs \( (N_1, N_2) \) that satisfy both conditions are:
- \( (0, 6) \)
- \( (4, 2) \)
- \( (6, 0) \)
- \( (8, 6) \)

### Conclusion
Thus, the values of \( N_1 \) and \( N_2 \) that make the number 774958N1N296 divisible by both 8 and 9 are:
- \( N_1 = 0 \) and \( N_2 = 6 \)
- \( N_1 = 4 \) and \( N_2 = 2 \)
- \( N_1 = 6 \) and \( N_2 = 0 \)
- \( N_1 = 8 \) and \( N_2 = 6 \)