We use the fact that if p is a factor of n then when n-m (0 <= m <= n) 
is divided by p < n-m, the remainder is p-m.

The required remainders are all 1 less than the required 
divisors respectively, so in this case m=1. We take the 
LCM of 9,8,6,5, and 4 which is 360 and subtract 1.  Since 
360 is a multiple of each of those, the remainders when
1 less that 360 or 359 is divided by them will be 1 less 
than them. 

Answer: 359

Checking
   39          44         59          71          89 
9)359       8)359      6)359       5)359       4)359
  27          32         30          35          32
   89          39         59           9          39
   81          32         54           5          36
    8           7          5           4           3
   
Edwin