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 Number = Power Remainder II (Posted on 2010-12-12)
(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 2x is divided by the base nine number 100. What are the possible values of x - if a remainder of x is obtained, whenever 5x 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 2y is divided by the base eleven number 100. What are the possible values y- if a remainder of y is obtained, whenever 5y is divided the base eleven number 100?

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

 No Solution Yet Submitted by K Sengupta No Rating

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 computer solution Comment 1 of 1
`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   nextOKrunbase 9; 2^x 34      3  7base 9; 5^x 58      6  4base 11; 2^x 65      5  10 104     9  5base 11; 5^x 45      4  1 64      5  9OK`

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

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