Turk J Math 30 (2006) , 1 – 2. \begin{matrix}c \\ ◯ \end{matrix} TÜBİTAK

Divisors of even numbers

Sophonsiba Beauchamp

Abstract

The greatest common divisor of the even numbers is 2. Key Words: greatest common divisor, even number

1. Introduction It is well known that, while only 0 is divisible by 0, every number is divisible by 1.

We show that every even number is divisible by 2, but by no greater number.

2. Preliminary results Let us recall:

Definition 2.1 A number is even if it is the sum of two equal numbers: Thus a is even, if a = b + c, where b = c. Lemma 2.2 Even numbers are divisible by 2. Proof. Let a be an even number. Then, by Definition 2.1, a = b + c, where b = c. Thus a = b + b = 1 · b + 1 · b = (1 + 1) · b = 2 · b. □

3. The main theorem Without further ado, let us proceed:

2000 AMS Mathematics Subject Classification: 53A, 53C.

BEAUCHAMP

Theorem 3.1 The even numbers have a greatest common divisor, which is 2. Proof. We have already established in Lemma 2.2 that 2 divides every even number. Suppose a > 2. We must show that there is an even number b that is indivisible by a. Let b = 2. Then the remainder of b after division by a is b itself, namely 2. But 2 ≠ 0. Therefore a does not divide b. But b was an even number. Therefore not every even number is divisible by a. But a was an arbitrary number greater than 2. Therefore the theorem is proved. □

We hope to prove in a forthcoming paper [1] that the greatest common divisor of the square numbers is 1. This is believed to have been known to the ancients [2], but the proof has been lost.

References

[1] S. Beauchamp, Divisors of squares, preprint.

[2] Philomathus of Ancyra, Thesaurus of Mathematics, lost manuscript.

Sophonsiba BEAUCHAMP Department of Philomathy Faulkner University Yoknapatawpha e-mail:sibbey@faulkner.edu Received November 22, 2006