Find the number of 6-digit numbers that can be found using the digits 1, 2, 3, 4, 5, 6 once such that the 6-digit number is divisible by its unit digit. (The unit digit is not 1).

Options

A. A) 620
B. B) 456
C. C) 520
D. D) 528

Answer

To find the number of 6-digit numbers that can be formed using the digits 1, 2, 3, 4, 5, and 6 exactly once, such that the number is divisible by its unit digit (with the restriction that the unit digit cannot be 1), we can follow these steps:

### Step 1: Identify Possible Unit Digits
The unit digit can be one of the following: 2, 3, 4, 5, or 6. We will analyze each case separately.

### Step 2: Case Analysis for Each Unit Digit

#### Case 1: Unit Digit = 2
- Any number ending in 2 is divisible by 2.
- The remaining digits are 1, 3, 4, 5, and 6.
- The number of arrangements of these 5 digits is \(5!\).
- Calculation: \(5! = 120\).

#### Case 2: Unit Digit = 3
- A number is divisible by 3 if the sum of its digits is divisible by 3.
- The sum of the digits 1, 2, 3, 4, 5, and 6 is 21, which is divisible by 3.
- The remaining digits are 1, 2, 4, 5, and 6.
- The number of arrangements of these 5 digits is \(5!\).
- Calculation: \(5! = 120\).

#### Case 3: Unit Digit = 4
- A number is divisible by 4 if the last two digits form a number that is divisible by 4.
- The possible pairs for the last two digits (with 4 as the unit digit) are 24 and 64.
- For each valid pair, we can arrange the remaining 4 digits.
- For the pair 24: Remaining digits are 1, 3, 4, 5, which can be arranged in \(4!\) ways.
- For the pair 64: Remaining digits are 1, 2, 3, 5, which can also be arranged in \(4!\) ways.
- Total arrangements for unit digit 4: \(4! + 4! = 24 + 24 = 48\).

#### Case 4: Unit Digit = 5
- Any number ending in 5 is divisible by 5.
- The remaining digits are 1, 2, 3, 4, and 6.
- The number of arrangements of these 5 digits is \(5!\).
- Calculation: \(5! = 120\).

#### Case 5: Unit Digit = 6
- Any number ending in 6 is divisible by 6 if it is even and the sum of its digits is divisible by 3.
- The remaining digits are 1, 2, 3, 4, and 5.
- The sum of these digits is 15, which is divisible by 3.
- The number of arrangements of these 5 digits is \(5!\).
- Calculation: \(5! = 120\).

### Step 3: Total Count of Valid Numbers
Now, we sum the valid arrangements from all cases:
- For unit digit 2: 120
- For unit digit 3: 120
- For unit digit 4: 48
- For unit digit 5: 120
- For unit digit 6: 120

Total = \(120 + 120 + 48 + 120 + 120 = 528\).

### Final Answer
The total number of 6-digit numbers that can be formed under the given conditions is **528**.

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