There exists a special six-digit number such that when this number is multiplied by four, its digits are reversed. Determine this special six-digit number. Note: digits can be used more than once.

Source: mathcontest.olemiss.edu 1/25/2010

**SOLUTION**

Let be the special six-digit number and be the reversed digit number. We make the following observations about .

**1**. The most significant digit of

*A*can only be {1, 2}, because if it is greater than or equal to 3, the multiplication by 4 will make

*A*a seven-digit number.

**2**. The most significant digit of *A* cannot be 1 because there is no way to produce the digit 1 by a multiplication by 4. Thus, the most significant digit of *A* must be 2.

**3**. If , then the most significant digit of is 8 because . Thus, .

**4**. Let . Then, . Since , we have a “carry” of 3. In order to get a zero digit out of a multiplication by 4 we have three possibilities:

But, it is imposssible to get 7, 17, or 27 out of a multiplication by 4.

**5**. Let . Then, . We have two choices for digit :

We have two cases to consider: .

**6.1 ****First case**: .

**6.1.0** Let . Then, . This time, the carry is equal to 1. In order to get a zero digit out of a multiplication by 4 we have three possibilities:

But, it is impossible to get 9, 19, or 29 out of a multiplication by 4.

**6.1.1** Let . Then . There is only one choice for digit :

**Check**

. Not an answer.

**6.1.2** Let . Then . In order to get a 2 digit out of a multiplication by 4 we have three possibilities:

But, it is impossible to obtain 11, 21, or 31 out of a multiplication by 4.

**6.1.3** Let . Then, . There are two choices for digit :

**Check**

. Not an answer.

. Not an answer.

**6.1.4** Let . Then, . In order to get a 4 digit out of a multiplication by 4 we have three possibilities:

But, it is impossible to get 13, 23, or 33 out of a multiplication by 4.

**6.1.5** Let . Then, . There are two choices for digit :

**Check**

. Not an answer.

. Not an answer.

**6.1.6**Let . Then, . In order to get a 6 digit out of a multiplication by 4 we have four possibilities:

**6.1.7**Let . Then, . There are two choices for digit :

**Check**

. Not an answer.

**6.1.8**Let . Then, . In order to get a digit 8 out of a multiplication by 4 we have three possibilities:

**6.1.9**Let . Then, . There are two choices for digit :

**Check**

. Not an answer.

. Not an answer.

**6.2**

**Second case**:

**6.2.0** Let . Then, . This time, the carry is equal to 3. In order to get a zero digit out of a multiplication by 4 we have three possibilities:

But, it is impossible to get 7, 17, or 27 out of a multiplication by 4.

**6.2.1** Let . Then, . There are two choices for digit :

**Check**

. Not an answer.

. Not an answer.

**6.2.2** Let . Then, . In order to get a 2 digit out of a multiplication by 4 we have three possibilities:

But, it is impossible to get 9, 19, or 29 out of a multiplication by 4.

**6.2.3** Let . Then, . There is only one choice for digit :

**Check**

. Not an answer.

**6.2.4** Let . Then, . In order to get a 4 digit out of a multiplication by 4 we have four possibilities:

But, it is impossible to get 11, 21, or 31 out of a multiplication by 4.

**6.2.5** Let . Then, . There are two choices for digit :

**Check**

. Not an answer.

. Not an answer.

**6.2.6** Let . Then, . In order to get a 6 digit out of a multiplication by 4 we have three possibilities:

But, it is impossible to get 13, 23, or 33 out of a multiplication by 4.

**6.2.7** Let . Then, . There are two choices for digit :

**Check**

. Not an answer.

. Not an answer.

**6.2.8** Let . Then, . In order to get an 8 digit out of a multiplication by 4 we have four possibilities:

But, it is impossible to get 5, 15, 25, or 35 out of a multiplication by 4.

**6.2.9** Let . Then, . There are two choices for digit :

**Check**

. Not an answer.

. YES! Finally!

**Answer**: 219978.