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Posers and Puzzles

Posers and Puzzles

  1. 04 Jul '08 10:48
    Is (19^92 - 91^29) / 90 a natural number or not?
    Motivate well...
  2. 04 Jul '08 12:03
    19^92 (mod 10)
    =-1^92 (mod 10)
    = 1 (mod 10
    19^92 (mod 9)
    =1 (mod 9)
    Therefore by CRT 19^92=1 (mod 90)
    Clearly 91^29=1 (mod 90), therefore the numerator is 0 (mod 90), yes it is an integer. Let me work out whether it is positive...
  3. 04 Jul '08 15:11
    Originally posted by FabianFnas
    Is [b](19^92 - 91^29) / 90 a natural number or not?
    Motivate well...[/b]
    We only need to know modulos for 2, 5, and 9, right? If all 3 are zero for the numerator, then we have a natural number.

    2 - Odd minus odd. ZERO
    5 - Cycles in 4, 4^4 - 1^1 = 15, ZERO
    9 - Cycles in 6, 1^2 - 1^5 = 0, ZERO

    Since all 3 are zero, the difference is divisible by 90..

    (Ignore this math) 3*92 = 276, 4.8*29 = 139.2, 276 - 48 = 228, estimate value has 22-23 digits in it and is positive..