Posted by William Bent on Monday, November 9, 2009 at 12:19pm.
Rather than use algebra, I would use trial and error. The number is between 7500 and 8499 and ends with zero. Possibilities that are also divisible by 6 are:
7500, 7530, 7560, 7590, 7620, 7630, 7680, 7710, 7740, 7770, 7800, 7830, 7860, 7890, 7920, 7950, 7980, 8010, 8040, 8070, 8100, 8130, 8160, 8190, 8220, 8250, 8280, 8310, 8340, 8370, 8400, 8430, and 8460.
The only ones that meet the last requirement -- about the hundreds and tens digits -- are 7630 and 7740. Only 7740 is not evenly divisible by 9; so that is the answer.