The tutor introduces Fermat’s Theorem with a first example. Fermat’s Theorem states that, for a prime number p and a number b not a multiple of p, bp-1 ≡ 1 (mod p). (See my post here for a working definition…

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The tutor introduces Fermat’s Theorem with a first example. Fermat’s Theorem states that, for a prime number p and a number b not a multiple of p, bp-1 ≡ 1 (mod p). (See my post here for a working definition…

Tagged with: Fermat's Theorem a^(p-1) ≡ 1 (mod p)

The tutor investigates a problem involving the remainder of a power. On page 48 of his Elementary Number Theory, second edition, Underwood Dudley requests the remainder when 20012001 is divided by 26. Solution: 2001 mod 26 = 25 ⇒ 2001…

The tutor investigates a problem involving composite numbers. For problem 4b, page 19, of his Elementary Number Theory (second edition), Dudley invites the reader to prove there are infinite n such that both 6n-1 and 6n+1 are composite. (Composite means…

The tutor solves a system of linear congruences. Back in my post from March 25, 2014, I explain that “mod” means remainder: for instance, 7 mod 3 = 1. Two numbers that, divided by a number n, give the same…

The tutor shows an interesting consequence of mod 3 arithmetic. Back in my March 25, 2014 post, I mentioned that mod means remainder. For example, 19 mod 4 = 3, because when you divide 19 by 4, you get 3…

The tutor tackles an age-old proof in a new (to him, anyway) manner. A famous theorem of number theory goes like this: For the integers a and b, there exists a solution with integers x and y to ax+by=1 if…

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