Number Pad Snake

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, 987456321÷9=109717369.{ \color{#EC7300}9}87456321 ÷ {\color{#EC7300}9} = 109717369. Will this method always result in an integer?

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