On this number pad, a nine-digit number is made by walking through each digit once, starting from a random digit and only moving to adjacent digits.
That number is then divided by its first digit. In this example,
$ \color{orange}987456321 ÷ \color{orange}9 = 109717369.$
Will this method always result in an integer?
My Solution:
Answer: No. (Without Calculation we can derive the result using divisibility test)
The given number pad has digits from 1 to 9. Clearly, it has 5 odd digits and 4 even digits. The next to even digits are odd & next to odd digits are even. The valid moves are odd to even (or) even to odd. If it start with even digit then it is not possible to end with even digit(since number pad has only 4 even digits). So it must start with odd digit, and the generated number will end with odd digit.
The possible start digits are 1, 3, 5, 7, or 9. Suppose, it start with 1, 3 or 9 then the generated number is divisible by 1, 3 or 9 respectively. Since all integers divisible by 1 and sum of digits 1+2+3+…+9 is 45, which is divisible by 3 and 9. The other two possible start digits are 5 or 7. If it start with 5 then the generated number not divisible by 5. Because it is not possible to get the end digit as 0 or 5. Hence the result always not an integer.
Looking forward other solutions