Integer Divisor Results/Integer Divides Zero

Theorem

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

Then:

$n \divides 0$

That is, $n$ divides $0$.


Proof

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

Therefore this result follows directly from Element of Integral Domain Divides Zero.

$\blacksquare$


Sources

  • 1969: C.R.J. Clapham: Introduction to Abstract Algebra ... (previous) ... (next): Chapter $3$: The Integers: $\S 10$. Divisibility
  • 1971: George E. Andrews: Number Theory ... (previous) ... (next): $\text {2-2}$ Divisibility: Example $\text {2-3}$
  • 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: P.M. Cohn: Algebra Volume 1 (2nd ed.) ... (previous) ... (next): Chapter $2$: Integers and natural numbers: $\S 2.2$: Divisibility and factorization in $\mathbf Z$
  • 1994: H.E. Rose: A Course in Number Theory (2nd ed.) ... (previous) ... (next): $1$ Divisibility: $1.1$ The Euclidean algorithm and unique factorization: $\text {(i)}$
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