A number in integer base n when written in base n1
becomes 1030n1
.The same number when
written in base n+1 becomes 444n+1.The original number written in base 10 is

• 1 * (n - 1)^3 + 3 * (n - 1) = 4 * (n + 1)^2 + 4 * (n + 1) + 4
  (n - 1) * ((n - 1)^2 + 3) = 4 * ((n + 1)^2 + (n + 1) + 1)
  (n - 1) * (n^2 - 2n + 1 + 3) = 4 * (n^2 + 2n + 1 + n + 2)
  (n - 1) * (n^2 - 2n + 4) = 4 * (n^2 + 3n + 3)
  n^3 - 2n^2 + 4n - n^2 + 2n - 4 = 4n^2 + 12n + 12
  n^3 - 2n^2 - n^2 - 4n^2 + 4n + 2n - 12n - 4 - 12 = 0
  n^3 - 7n^2 - 6n - 16 =0
  n = 8
  
  1 * (8 - 1)^3 + 3 * (8 - 1) =>
  1 * 7^3 + 3 * 7 =>
  343 + 21 =>
  364
• 8 years agoHelpfull: Yes(8) No(1)
• let n=8;
  when base is n+1
  364%9 =4
  40%9=4
  4%9=4
  444 base n+1
  
  when base is n-1
  364%7=0
  52%7=3
  7%7=0
  1%7=1
  1030 base n-1
  
  
  
• 8 years agoHelpfull: Yes(4) No(13)