Close Divisors

Find four 3digit numbers which have the same first digit, such that
the sum of all four numbers is divisible by three of them.

For each n (4<=n<=7), can you find a set of n+1 integers, all
the same length and beginning with the same digit, such that their sum
is divisible by n of them? Try to minimize the sum.

For each n (2<=n<=7),
find the smallest integer M, such that when you divide M evenly by
k, k+1, ..., k+n1, you get n integers, all beginning with the same
digit. You can choose k in each case.
Source: Recent puzzle reading (still looking for the actual source.)
Solution
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