Prove that for every 𝒏∈ℕ, 𝟑^𝟔𝒏−𝟐^𝟔𝒏 is divisible by 35 | Moscow Mathematical Olympiad Problem

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We apply a special formula to complete the proof that for every positive integer n, 3^6n - 2^6n is divisible by 35.

  1. Dr. Wang
  2. Moscow Mathematical Olympiad
  3. USSR Olympiad Question
  4. Number Theory
  5. divisible by
  6. n is divisible by
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One could also use the identity for the difference of powers with equal exponents right from the start: x^n - y^n = (x - y)(x^(n-1) + x^(n-2)*y + …). Setting x = 3^6 and y = 2^6, one immediately obtains x - y = 665 wich is divisible by 35.
Of couse, it would also be a neat excersise to proove the statement by induction 😉

florianbuerzle
Автор

I completed this induction proof rather quickly, although my high school class has this form of induction and the only other type is induction proof of sums.

kylehurley