Integer Divisor Results/One Divides all Integers

Theorem

Let $n \in \Z$ be an integer.

Then:

$\ds 1$

$\divides$

$\ds n$

$\ds -1$

$\divides$

$\ds n$

where $\divides$ denotes divisibility.

From Integers form Integral Domain, the concept divisibility is fully applicable to the integers.

Therefore this result follows directly from Unity Divides All Elements.

$\blacksquare$

1971: George E. Andrews: Number Theory ... (previous) ... (next): $\text {2-2}$ Divisibility: Example $\text {2-2}$
1971: Allan Clark: Elements of Abstract Algebra ... (previous) ... (next): Chapter $1$: Properties of the Natural Numbers: $\S 22$
1979: G.H. Hardy and E.M. Wright: An Introduction to the Theory of Numbers (5th ed.) ... (previous) ... (next): $\text I$: The Series of Primes: $1.1$ Divisibility of integers
1980: David M. Burton: Elementary Number Theory (revised ed.) ... (previous) ... (next): Chapter $2$: Divisibility Theory in the Integers: $2.2$ The Greatest Common Divisor: Theorem $2 \text{-} 2 \ (1)$
1982: Martin Davis: Computability and Unsolvability (2nd ed.) ... (previous) ... (next): Appendix $1$: Some Results from the Elementary Theory of Numbers
1983: François Le Lionnais and Jean Brette: Les Nombres Remarquables ... (previous) ... (next): $1$
1994: H.E. Rose: A Course in Number Theory (2nd ed.) ... (previous) ... (next): $1$ Divisibility: $1.1$ The Euclidean algorithm and unique factorization: $\text {(i)}$