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Determine divisibility

  1. Jul 12, 2016 #1
    • Member warned that the homework template is not optional
    Hi I'm reading a text about modular arithmetic,

    Prove that 16^43 - 10^26 actually is divisible by 21.
    They separate it by showing it is divisible by 7 and 3

    they showed [tex] 16 \equiv 2 \textrm{ mod 7} \\
    16^2 \equiv 2^2 \equiv 4 \textrm{ mod 7} \\
    16 \equiv 2^3 \equiv 1 \textrm{ mod 7} \\ [/tex]
    So there is a pattern of length 3.

    They later made 43 = 3 * 14 +1 . so,
    [tex]
    16^{43} \equiv 16^1 \equiv 2 \textrm{ mod 7} \\ [/tex]

    whats the reasoning with 43 = 3 * 14 + 1 ?
     
  2. jcsd
  3. Jul 13, 2016 #2

    ehild

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    ##16^{43}=16^{42}\cdot 16 = (16^3)^{14}\cdot16##. You know that 16^3= 1 mod 7, 16=2 mod 7.
     
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