Unique Quotient in Natural Numbers

Definition

Let $\N$ be the natural numbers.

Let $n \in \N$ and $m \in \N_{>0}$ such that:

$m \divides n$

where $m \divides n$ denotes that $m$ is a divisor of $n$.

Then there exists exactly one element $p \in \N$ such that $m \times p = n$.

Proof

Let $n = m \times p$.

Such a $p$ exists because $m$ is a divisor of $n$.

Suppose that $n = 0$.

Then from Natural Numbers have No Proper Zero Divisors it follows that $p = 0$.

Thus in this case the unique value of $p$ is zero.

Now suppose $n \ne 0$.

Let $n = m \times p = m \times q$ for $p, q \in \N$.

Then from Natural Number Multiplication is Cancellable it follows that $p = q$.

Hence the result.

$\blacksquare$

Sources

• 1965: Seth Warner: Modern Algebra ... (previous) ... (next): Chapter $\text {III}$: The Natural Numbers: $\S 16$: The Natural Numbers