Ponder This

It’s the beginning of the month and the solution to last month’s Ponder This challenge is up, as well as the puzzle for August:

For K as large as possible, produce a K-digit integer M such that for each N=1,2,…,K, the integer given by the first N digits of M is divisible by N.
An example is K=4, M=7084, because 7 is divisible by 1; 70 is divisible by 2; 708 is divisible by 3; and 7084 is divisible by 4.

I guess that the largest K is around 28.

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