February 2013

# Solution

Okay, the digits which can be read upside down are 0,1,6,8 and 9. Thus, the number comprised these digits.Suppose the number was a four-digit one, say 6890, then when turned upside down it will appear as 0689. Take note of this.

Now since the upside-down number was greater than the actual one by 78633, this implies that it should start either with 8 or 9 (any other number in this place cannot give a difference of 78633). This means that the last digit in the actual number should be either 8 (it will appear as 8 when upside down) or 6 (this will appear as 9 when upside down).

The upside down number is greater than the actual by 78633. Consider the last digit, 3. Now considering the five digits 0, 1, 6, 8 and 9, we can get 3 only when we have 1-8 (this will become 11-8 after the necessary borrow) or 9-6. Therefore, the first digit in the actual number can be either 9 or 1. Thus, we have concluded that the first digit of the actual number can be either 9 or 1 and the last digit can be either 8 or 6. Thus, the first and last digit combinations could be (9, 8), (9, 6), (1, 8) or (1, 6)

Let us consider the (9, 8) combination. The other three digits (0, 1, 6) can have six different arrangements among them - 106, 160, 016, 061, 610 and 601. So the five-digit number after considering these combinations could be: 91068, 91608, 90168, 90618, 96108 and 96018. Any of these numbers when turned upside down will begin with 8 and the resulting number will be smaller (for example: 91068 will become 89016 which is smaller). So the entire series of numbers with the (9, 8) combination can be ruled out.

Let us consider the (9, 6) combination. Here the other three digits - 0, 1 and 8 - will have six arrangements and, accordingly, six numbers will be obtained. These are: 91086, 91806, 90186, 90816, 98106 and 98016. All these numbers when turned upside down will also begin with 9 (for example: 91086 will become 98016. Therefore, their difference cannot be 78633 under any circumstances. This series too can, therefore, be ruled out.

This leaves us with the (1, 6) and (1, 8) combinations. The five-digit numbers that we get from these combinations are: 10896, 18096, 10986, 19086, 18906, 19806, 10968, 19068, 10698, 16098, 19608 and 16908. These are only twelve numbers and so it would not be difficult to turn each one upside down to see whether the difference is 78633. The number 10968 when turned upside down becomes 89601 and the difference is, in fact, 78633. The actual number, therefore, is 10968.

• Union Public Service Commission - www.upsc.gov.in
• IIT-Kharagpur - www.iitkgp.ac.in
• Indian Statistical Institute - www.isical.ac.in
• Indian Institute of Technology Madras - www.iitm.ac.in
• Indian Institute of Management, Ahmedabad - www.iimahd.ernet.in
• Indian Institute of Mass Commission - www.iimc.nic.in
• IIT Bombay - www.iitb.ac.in