(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.
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).
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Posted by Charlie
on 2010-12-13 02:29:00 |