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Positively congruent to a T (Posted on 2008-10-30) Difficulty: 3 of 5
Determine the total number of positive integers T ≥ 2, for which the relationship: Y25 ≡ Y (mod T) holds true for all possible values of a positive integer Y ≥ 2, where the "≡" symbol denotes congruence.

No Solution Yet Submitted by K Sengupta    
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Solution computer solution (spoiler) Comment 1 of 1

As 2^25 = 33,554,432, T greater than this value need not be tested as those modular results for this power of 2 would not reduce it to 2.

The following program tests T up to 34,000,000. (Variable I was used in place of Y):

 10   for T=2 to 34000000
 20    Good=1
 30    for I=1 to T-1
 40       if (I^25)@T<>I then Good=0:cancel for:goto 100
 50    next
100    if Good then print T;:Ct=Ct+1
150   next T
200   print:print Ct

It found 31 values of T:

2  3  5  6  7  10  13  14  15  21  26  30  35  39  42  65  70  78  91  105  130 182  195  210  273  390  455  546  910  1365  2730


  Posted by Charlie on 2008-10-30 11:10:06
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