To solve the problem of finding the number of different 5-digit numbers divisible by 4 using the digits 4, 5, 6, 7, 8, and 9 without repetition, we can follow these steps:
### Step 1: Understand the divisibility rule for 4
A number is divisible by 4 if the number formed by its last two digits is divisible by 4. Therefore, we need to focus on the last two digits of our 5-digit number.
### Step 2: Identify possible pairs of last two digits
We will check which pairs of the given digits (4, 5, 6, 7, 8, 9) form numbers that are divisible by 4. The possible pairs are:
- 44 (not allowed, repetition)
- 45 (not divisible by 4)
- 46 (not divisible by 4)
- 47 (not divisible by 4)
- 48 (divisible by 4)
- 49 (not divisible by 4)
- 54 (not divisible by 4)
- 56 (divisible by 4)
- 57 (not divisible by 4)
- 58 (not divisible by 4)
- 64 (divisible by 4)
- 65 (not divisible by 4)
- 67 (not divisible by 4)
- 68 (divisible by 4)
- 69 (not divisible by 4)
- 74 (not divisible by 4)
- 75 (not divisible by 4)
- 76 (divisible by 4)
- 78 (not divisible by 4)
- 84 (divisible by 4)
- 85 (not divisible by 4)
- 86 (not divisible by 4)
- 87 (not divisible by 4)
- 94 (not divisible by 4)
- 95 (not divisible by 4)
- 96 (divisible by 4)
Thus, the valid pairs of last two digits are: **48, 56, 64, 68, 76, 84, 96**.
### Step 3: Calculate the number of valid 5-digit numbers for each pair
For each valid pair of last two digits, we will determine how many different 5-digit numbers can be formed using the remaining digits.
1. **Last two digits: 48**
- Remaining digits: 5, 6, 7, 9 (4 choices)
- Number of arrangements: 4! = 24
2. **Last two digits: 56**
- Remaining digits: 4, 7, 8, 9 (4 choices)
- Number of arrangements: 4! = 24
3. **Last two digits: 64**
- Remaining digits: 5, 7, 8, 9 (4 choices)
- Number of arrangements: 4! = 24
4. **Last two digits: 68**
- Remaining digits: 4, 5, 7, 9 (4 choices)
- Number of arrangements: 4! = 24
5. **Last two digits: 76**
- Remaining digits: 4, 5, 8, 9 (4 choices)
- Number of arrangements: 4! = 24
6. **Last two digits: 84**
- Remaining digits: 5, 6, 7, 9 (4 choices)
- Number of arrangements: 4! = 24
7. **Last two digits: 96**
- Remaining digits: 4, 5, 7, 8 (4 choices)
- Number of arrangements: 4! = 24
### Step 4: Total the arrangements
Now, we sum the arrangements from all valid pairs:
- Total = 24 + 24 + 24 + 24 + 24 + 24 + 24 = 168
### Conclusion
The total number of different 5-digit numbers that can be formed using the digits 4, 5, 6, 7, 8, and 9, which are divisible by 4, is **168**.
### Final Answer
The value of \( n \) is **168** (Option c).
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