Question

For a rational number r, its period is the length of the smallest repeating block in its decimal expansion. For example, the number r = 0.123123123… has period 3. If S denotes the set of all rational number r of the form r = bar 0.abcdefgh having period 8, find the sum of all the elements of S.

Answer 1

X = 0.abcdefghabcdefgh……………..

If unit digit is 1 then the next three places can be filled in 103 ways hence sum of all the digits at unit place 

It also include numbers with period 1 and period 2

So sum of the numbers with period 8 = Total – Sum of the numbers with period 1, 2, 4

= 5 × 107 – 5 × 103 = 49995000