(A) Determine all possible value(s) of a 2-digit non leading zero base nine positive integer x such that we will obtain a remainder of x, whenever 2^x is divided by the base nine number 100. What are the possible values of x - if a remainder of x is obtained, whenever 5^x is divided the base nine number 100?

(B) Determine all possible value(s) of a 2-digit non leading zero base eleven positive integer y such that we will obtain a remainder of y, whenever 2^y is divided by the base eleven number 100. What are the possible values y- if a remainder of y is obtained, whenever 5^y is divided the base eleven number 100?

*** For an extra challenge, solve this problem without using a computer program.

list
    5   print "base 9; 2^x"
   10   for X=9 to 80
   20      V=(2^X)@81
   25      if V=X then print X,X\9;X@9
   30   next
   35   print "base 9; 5^x"
   40   for X=9 to 80
   50      V=(5^X)@81
   55      if V=X then print X,X\9;X@9
   60   next
  100   print "base 11; 2^x"
  110   for X=10 to 120
  120      V=(2^X)@121
  125      if V=X then print X,X\11;X@11
  130   next
  135   print "base 11; 5^x"
  140   for X=10 to 120
  150      V=(5^X)@121
  155      if V=X then print X,X\11;X@11
  160   next
OK
run
base 9; 2^x
 34      3  7
base 9; 5^x
 58      6  4
base 11; 2^x
 65      5  10
 104     9  5
base 11; 5^x
 45      4  1
 64      5  9
OK

So, in base 9, the numbers are 37 (decimal 34) and 64 (decimal 58) for the 2^x and 5^x cases respectively.

For base 11, the 2^x case has the solutions 5A (decimal 65) and 95 (decimal 104), while the 5^x case has the solutions 41 (decimal 45) and 59 (decimal 64).

Posted by Charlie on 2010-12-13 02:29:00