Given that 2^2004 is a 604-digit number whose first digit is 1, how many elements of the set S = {2^0,2^1,2^3...,2^2003} have a first digit of 4?

• 195 elements of the set S = {2^0,2^1,2^3...,2^2003} have a first digit of 4.
• 12 years agoHelpfull: Yes(2) No(0)
• has digits and starts with a , there are numbers of digits for the powers of from to .
  
  
  so
  
  where is the number of numbers of digits that have powers of , and is the number of numbers of digits that have powers of . We're looking for .
  
  Solving for , the answer is 195
• 12 years agoHelpfull: Yes(1) No(0)
• this problem is from book 104 Number Theory Problems: From the Training of the USA IMO Team
  
  i explained answer in details in below
  
  The smallest power of 2 with a given number of digits has a first digit of 1, and there are elements of S with n digits for each integer n
• 8 years agoHelpfull: Yes(0) No(0)
• this problem is from book 104 Number Theory Problems: From the Training of the USA IMO Team
  
  i explained answer in details in below
  
  The smallest power of 2 with a given number of digits has a first digit of 1, and there are elements of S with n digits for each integer n
• 8 years agoHelpfull: Yes(0) No(0)