The LCM of 6,12,18,24,30 is 360, so the only positive integers that
will leave remainder 4 when divided by all of those is 4 more than
a multiple of 360.  Such a number would be of the form 360n+4.

  1000 ≦ 360n+4 ≦ 9999

Subtract 4 from all three sides:

    996 ≦ 360n ≦ 9995

Divide all three sides by 360:

2.7666... ≦ n ≦ 27.763888... 

Since n is an integer:

       3 ≦ n ≦ 27

So the smallest such 4 digit number is when n=3, 360*3+4 = 1084.
And the greatest such 4 digit number is when n=27, 360*27+4 = 9724.

Answer: 9724.

Edwin