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63 Division Decision (Posted on 2016-11-10) Difficulty: 3 of 5
N is a positive integer which is expressible as the sum of cubes of two positive integers.
Given that N is not divisible by 9, find the possible remainders when N is divided by 63.

No Solution Yet Submitted by K Sengupta    
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Solution computer solution | Comment 1 of 3
All sums of two cubes a^3 + b^3 where a+b totaled no more than 10,000 were tested. Remainders, r, all were found with totals a+b summing no more than 23 (plus additional times at higher values, of course).

   lowest
 r   N        cubes          roots

 1 1072      343  729        7    9
 2    2        1    1        1    1
 7  133        8  125        2    5
 8 9269        8 9261        2   21
16   16        8    8        2    2
19 1027       27 1000        3   10
20 1343      343 1000        7   10
26  152       27  125        3    5
28   28        1   27        1    3
29  344        1  343        1    7
34 2869      125 2744        5   14
35   35        8   27        2    3
37  730        1  729        1    9
43 2752        8 2744        2   14
44  737        8  729        2    9
47 2000     1000 1000       10   10
55  370       27  343        3    7
56 1001        1 1000        1   10
61  250      125  125        5    5
62 2771       27 2744        3   14

from

DefDbl A-Z
Dim crlf$, had(62, 2)


Private Sub Form_Load()
 Form1.Visible = True
 
 
 Text1.Text = ""
 crlf = Chr$(13) + Chr$(10)
 
 For tot = 2 To 10000
   For a = 1 To tot / 2
     b = tot - a
     n = a * a * a + b * b * b
     q = Int(n / 9)
     r = n - 9 * q
     If r > 0 Then
       q = Int(n / 63)
       r = n - 63 * q
       If had(r, 0) = 0 Then
         had(r, 0) = n: had(r, 1) = a: had(r, 2) = b
       End If
     End If
   Next
 Next
 
 For i = 0 To 62
  If had(i, 0) Then
  Text1.Text = Text1.Text & mform(i, "##") & mform(had(i, 0), "####0") & "    "
  Text1.Text = Text1.Text & mform(had(i, 1) ^ 3, "####0") & mform(had(i, 2) ^ 3, "####0") & "    "
  Text1.Text = Text1.Text & mform(had(i, 1), "####0") & mform(had(i, 2), "####0") & crlf
  End If
 Next
 
 Text1.Text = Text1.Text & crlf & crlf & " done"
  
End Sub

Function mform$(x, t$)
  a$ = Format$(x, t$)
  If Len(a$) < Len(t$) Then a$ = Space$(Len(t$) - Len(a$)) & a$
  mform$ = a$
End Function

If a were not allowed to equal b the list would have the same set of remainders

 1 1072      343  729        7    9
 2   65        1   64        1    4
 7  133        8  125        2    5
 8 9269        8 9261        2   21
16  520        8  512        2    8
19 1027       27 1000        3   10
20 1343      343 1000        7   10
26  152       27  125        3    5
28   28        1   27        1    3
29  344        1  343        1    7
34 2869      125 2744        5   14
35   35        8   27        2    3
37  730        1  729        1    9
43 2752        8 2744        2   14
44  737        8  729        2    9
47 3197     1000 2197       10   13
55  370       27  343        3    7
56 1001        1 1000        1   10
61 5038      125 4913        5   17
62 2771       27 2744        3   14

  Posted by Charlie on 2016-11-10 12:42:32
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