x is a five digit number. The digit in ten thousands place is 1. The number formed by its digits in units and tens places is divisible by 4. The sum of all the digits is divisible by 3. If 5 and 7 also divide x, then x will be :

Options

A. 14020
B. 12060
C. 10020
D. 10080

Answer

To solve the problem step-by-step, we need to find a five-digit number \( x \) that satisfies the following conditions:

1. The digit in the ten-thousands place is 1.
2. The number formed by the digits in the units and tens places is divisible by 4.
3. The sum of all the digits is divisible by 3.
4. The number \( x \) is divisible by both 5 and 7.

### Step 1: Determine the structure of the number
Since \( x \) is a five-digit number and the digit in the ten-thousands place is 1, we can express \( x \) as:
\[ x = 1abcd \]
where \( a, b, c, d \) are the digits in the thousands, hundreds, tens, and units places respectively.

### Step 2: Check the divisibility by 4
The number formed by the digits in the units and tens places (i.e., \( cd \)) must be divisible by 4. The possible two-digit combinations for \( cd \) that are divisible by 4 include:
- 00, 04, 08, 12, 16, 20, 24, 28, 32, 36, 40, 44, 48, 52, 56, 60, 64, 68, 72, 76, 80, 84, 88, 92, 96

### Step 3: Check the sum of the digits
The sum of the digits \( 1 + a + b + c + d \) must be divisible by 3.

### Step 4: Check divisibility by 5
For \( x \) to be divisible by 5, \( d \) must be either 0 or 5. Since \( cd \) must also be divisible by 4, we can only use \( d = 0 \) (as 5 cannot form a two-digit number with \( c \) that is divisible by 4).

### Step 5: Possible values for \( c \)
Since \( d = 0 \), we need \( c0 \) to be divisible by 4. The valid values for \( c \) that make \( c0 \) divisible by 4 are:
- 0, 2, 4, 6, 8

### Step 6: Check divisibility by 7
We will also need to check which of the candidates formed by \( 1abc0 \) is divisible by 7.

### Step 7: Testing candidates
Let’s test the combinations:

1. **For \( c = 0 \)**:
- \( x = 1ab00 \)
- Sum of digits: \( 1 + a + b + 0 + 0 = 1 + a + b \) (check for divisibility by 3)
- Check divisibility by 7.

2. **For \( c = 2 \)**:
- \( x = 1ab20 \)
- Sum of digits: \( 1 + a + b + 2 + 0 = 3 + a + b \) (check for divisibility by 3)
- Check divisibility by 7.

3. **For \( c = 4 \)**:
- \( x = 1ab40 \)
- Sum of digits: \( 1 + a + b + 4 + 0 = 5 + a + b \) (check for divisibility by 3)
- Check divisibility by 7.

4. **For \( c = 6 \)**:
- \( x = 1ab60 \)
- Sum of digits: \( 1 + a + b + 6 + 0 = 7 + a + b \) (check for divisibility by 3)
- Check divisibility by 7.

5. **For \( c = 8 \)**:
- \( x = 1ab80 \)
- Sum of digits: \( 1 + a + b + 8 + 0 = 9 + a + b \) (check for divisibility by 3)
- Check divisibility by 7.

### Step 8: Final checks
After checking each of these combinations, we find that:
- \( 10080 \) is divisible by 7, and the sum of its digits \( 1 + 0 + 0 + 8 + 0 = 9 \) is divisible by 3.

Thus, the value of \( x \) that satisfies all the conditions is:
\[ \boxed{10080} \]