Question

find how many 3 digits numbers satisfy all the following conditions
if it is divided by 2, the remainder is 1,
if it is divided by 3, the remainder is 2,
if it is divided by 4, the remainder is 3,
if it is divided by 5, the remainder is 4,
if it is divided by 8, the remainder is 7,

Answer 1

let the number N satisfy the conditions.
Let x, y, z, w, v be some integers.  

N can be written as 2 x + 1 or  2 x - 1.  so let us write N as 
            
         N = 2 x - 1
         N = 3 y - 1            or   3 x' + 2      where x' = y -1
         N = 4 z - 1           or    4 z' + 3     where z' = z - 1
         N = 5 w - 1          or    5 w' + 4
         N = 8 v - 1          or    8 v' + 7     
       
  So N + 1 = 2 x = 3 y = 4 z = 5 w = 8 v

         N+1 is a multiple of 2,3,4,5, and 8

       N+1 is a multiple of  lcm of them.
   
           LCM  =  2*3*2*5*2 = 120

       Three digit multiples of 120 are 120, 240, 360, 480, 600, 720, 840, 960

      N = 119, 239, 359, 479, 599 , 719, 839, 959

There are eight of them.