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 774958 `N_1`96`N_2` so that it is divisible by both 8 and 9, we will follow these steps:

### Step 1: Check divisibility by 9
A number is divisible by 9 if the sum of its digits is divisible by 9.

The digits of the number are:
- 7, 7, 4, 9, 5, 8, \( N_1 \), 9, 6, \( N_2 \)

Calculating the sum of the known digits:
\[ 7 + 7 + 4 + 9 + 5 + 8 + 9 + 6 = 55 \]
Including \( N_1 \) and \( N_2 \):
\[ \text{Sum} = 55 + N_1 + N_2 \]

For the number to be divisible by 9, \( 55 + N_1 + N_2 \) must be divisible by 9.

### Step 2: Determine the minimum value for \( N_1 + N_2 \)
To find the smallest multiple of 9 greater than 55, we calculate:
\[ \text{Next multiple of 9 after 55} = 63 \]
Thus,
\[ 55 + N_1 + N_2 = 63 \implies N_1 + N_2 = 8 \]
Alternatively, the next multiple of 9 would be 72:
\[ 55 + N_1 + N_2 = 72 \implies N_1 + N_2 = 17 \]
However, since \( N_1 \) and \( N_2 \) are single digits (0-9), the only feasible equation is:
\[ N_1 + N_2 = 8 \]

### Step 3: Check divisibility by 8
A number is divisible by 8 if the number formed by its last three digits is divisible by 8. The last three digits of our number are `96N_2`.

We will check the values of \( N_2 \) from 0 to 9 to find which makes `96N_2` divisible by 8.

- For \( N_2 = 0 \): 960 ÷ 8 = 120 (divisible)
- For \( N_2 = 1 \): 961 ÷ 8 = 120.125 (not divisible)
- For \( N_2 = 2 \): 962 ÷ 8 = 120.25 (not divisible)
- For \( N_2 = 3 \): 963 ÷ 8 = 120.375 (not divisible)
- For \( N_2 = 4 \): 964 ÷ 8 = 120.5 (not divisible)
- For \( N_2 = 5 \): 965 ÷ 8 = 120.625 (not divisible)
- For \( N_2 = 6 \): 966 ÷ 8 = 120.75 (not divisible)
- For \( N_2 = 7 \): 967 ÷ 8 = 120.875 (not divisible)
- For \( N_2 = 8 \): 968 ÷ 8 = 121 (divisible)
- For \( N_2 = 9 \): 969 ÷ 8 = 121.125 (not divisible)

### Step 4: Find valid pairs for \( N_1 \) and \( N_2 \)
From our calculations:
- If \( N_2 = 0 \), then \( N_1 = 8 \) (valid)
- If \( N_2 = 8 \), then \( N_1 = 0 \) (valid)

Thus, the valid pairs \( (N_1, N_2) \) are:
1. \( (8, 0) \)
2. \( (0, 8) \)

### Conclusion
The values of \( N_1 \) and \( N_2 \) that make the number 774958 `N_1`96`N_2` divisible by both 8 and 9 are:
- \( N_1 = 0 \) and \( N_2 = 8 \) or \( N_1 = 8 \) and \( N_2 = 0 \).