The problem states that Pat took ' $N$ ' whole days to complete the project.
This means the total number of hours he took must be a multiple of 24 (since there are 24 hours in a day, and N is a whole number of days).

We need to check which of the given options, when the missing digit is filled, can be a multiple of 24.

For a number to be a multiple of 24 , it must be divisible by both 3 and 8 (since $\(24=3 \times 8\)$ ).

Divisibility Rules:
- Divisibility by 3: The sum of the digits must be divisible by 3 .
- Divisibility by 8: The number formed by the last three digits must be divisible by 8.

Let's check each option:
(A) 6_98
- Divisibility by 8: The last three digits form '_98'.
- If _ is $\(0,098 / 8\)$ is not an integer.
- If _ is $\(1,198 / 8\)$ is not an integer.
- If _ is $\(2,298 / 8\)$ is not an integer.
- If _ is $\(3,398 / 8\)$ is not an integer.
- If _ is $\(4,498 / 8\)$ is not an integer.
- If _ is $\(5,598 / 8\)$ is not an integer.
- If _ is $\(6,698 / 8\)$ is not an integer.
- If _ is \(7, 798/8\) is not an integer.
- If _ is $\(8,898 / 8\)$ is not an integer.
- If _ is $\(9,998 / 8\)$ is not an integer.

- Alternatively, for a number to be divisible by 8 , it must be even (which 6_98 is not, as 98 is even, but the hundreds digit makes the whole number potentially not divisible by 8 unless the last two digits are divisible by 4 , and last three by 8 ). A quick check: Any number ending in _98 is an even number, but for divisibility by 8, the last three digits must form a number divisible by 8. No _98 will be divisible by 8. For example, $\(100 / 8=12.5\)$. $\(98 / 8\)$ gives remainder. The closest multiples of 8 are 96,104 . So 98 won't work. The hundreds digit would need to make it work. For example, 198 is not divisible by 8. 298 is not. This option is highly unlikely.
(B) 6_36
- Divisibility by 8: The last three digits form ' 36 '.
- We need _36 to be divisible by 8 .
- Test digits for _:
- 036/8 (not divisible)
- $136 / 8=17$ (divisible!) So, 6136 is a possibility.
- 236/8 (not divisible)
- $336 / 8=42$ (divisible!) So, 6336 is a possibility.
- 436/8 (not divisible)
- $536 / 8=67$ (divisible!) So, 6536 is a possibility.
- 636/8 (not divisible)

- $736 / 8=92$ (divisible!) So, 6736 is a possibility.
- 836/8 (not divisible)
- $936 / 8=117$ (divisible!) So, 6936 is a possibility.
- Since we found cases where it's divisible by 8 , now check Divisibility by 3 : The sum of digits $6+$ _ $+3+6=15+$ _.
- We need 15 + _ to be divisible by 3 . This means _ can be $0,3,6,9$.
- From divisibility by 8 , we found _ can be $1,3,5,7,9$.
- The common digits for _ are 3 and 9.
- If _ = 3, sum is 18 (divisible by 3), last three digits 336 (divisible by 8). So 6336 is a multiple of 24.
- If _ = 9, sum is 24 (divisible by 3), last three digits 936 (divisible by 8). So 6936 is a multiple of 24.
- Since we found a possible number (e.g., 6336 or 6936), this option could be the number of hours.
(C) 6_50
- Divisibility by 8: A number ending in 50 is not divisible by 8. For a number to be divisible by 8, its last three digits must form a multiple of 8 . No number ending in 50 will be divisible by 8 (multiples of 8 usually end in $00,08,16,24$, etc. - specifically, 50 is not divisible by 8 , and an odd hundreds digit will lead to odd tens if we were thinking about 250,450 , etc. Only even hundreds digits yield multiples of 8 at the end: $\(100 / 8=12.5,200 / 8=25,300 / 8=37.5\)$, 400/8=50.
- Check: _50. $\(150 / 8=18\)$ R 6. $\(250 / 8=31\)$ R 2. $\(350 / 8=43\)$ R 6. No multiple of 8 ends in 50.
- This option cannot be a multiple of 24 .

(C) 6_50
- Divisibility by 8: A number ending in 50 is not divisible by 8 . For a number to be divisible by 8, its last three digits must form a multiple of 8 . No number ending in 50 will be divisible by 8 (multiples of 8 usually end in $00,08,16,24$, etc. - specifically, 50 is not divisible by 8 , and an odd hundreds digit will lead to odd tens if we were thinking about 250, 450, etc. Only even hundreds digits yield multiples of 8 at the end: 100/8=12.5, 200/8=25, 300/8=37.5, 400/8=50.
- Check: _50. $150 / 8=18$ R 6. $250 / 8=31$ R 2. $350 / 8=43$ R 6. No multiple of 8 ends in 50 .
- This option cannot be a multiple of 24.
(D) $6 \_90$
- Divisibility by 8 : A number ending in 90 is not divisible by 8 .
- Check: _90. 090/8 (not divisible). 190/8 (not divisible). No number ending in _90 will be divisible by 8 .
- This option cannot be a multiple of 24.
(E) 8_62
- Divisibility by 8 : A number ending in 62 is not divisible by 8 .
- Check: _62. 062/8 (not divisible). 162/8 (not divisible). No number ending in _62 will be divisible by 8 .
- This option cannot be a multiple of 24.

Therefore, the only option that could be the number of hours Pat took to complete the project is (B).

The final answer is B.