Unique Quotient in Natural Numbers
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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