Answer to Question #54501 in Discrete Mathematics for Mike

Question

Using Euler’s theorem, explain why it is possible to pass through all of the stations by traversing every rail only once.

Answer 1

In number theory, Euler's theorem may be used to easily reduce large powers modulo n. For example, consider finding the ones place decimal digit of 7222, i.e. 7222 (mod 10). Note that 7 and 10 are coprime, and φ(10) = 4. So Euler's theorem yields 74 ≡ 1 (mod 10), and we get 7222 ≡ 74 × 55 + 2 ≡ (74)55 × 72 ≡ 155 × 72 ≡ 49 ≡ 9 (mod 10).
Euler's theorem also forms the basis of the RSA encryption system, where the net result of first encrypting a plaintext message, then later decrypting it, amounts to exponentiating a large input number by kφ(n) + 1, for some positive integer k. Euler's theorem then guarantees that the decrypted output number is equal to the original input number, giving back the plaintext.