Basic Divisibility Theory

Definition 53   We say that $a$ divides $b$, written $a\vert b$, if there is a natural number $n$ with $an=b$.

Note that $a\vert b$ is a sentence, not a number.

With this definition we can provide proofs of the following:

Proposition 83   If $a\vert b$ and $b\vert c$ then $a\vert c$.

(2 Points )

Proposition 84   If $a\vert b$ and $a\vert c$ then $a\vert b+c$.

(2 Points )

Proposition 85   For any $a$, $a\vert a$.

(2 Points )

Proposition 86   For any $a$, $1\vert a$.

(2 Points )

Proposition 87   For any $a$, $a\vert$.

(2 Points )

Proposition 88   If $a\vert b$ and $b\in \ensuremath{\mathbb{N}}^+$ then $a\leq b$.

(3 Points )

Proposition 89   If $a\vert b$ and $a\vert b+c$ then $a\vert c$.

(4 Points )

Problem 90   Find counterexamples for the following:

1. If $a\vert bc$ then either $a\vert b$ or $a\vert c$.
2. Either $a\vert b$ or $b\vert a$.
3. If $a\vert c$ and $b\vert c$ then $ab\vert c$.
4. If $a\vert c$ and $b\vert c$ then $a+b\vert c$.

(4 Points )

Definition 54   A prime number $p$ is one with the property that exactly two natural numbers divide $p$, namely 1 and $p$.

Proposition 91   Neither 0 nor 1 is prime.

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Proposition 92   For any number $a$ and any prime $p$ either $p\vert a$ or if $n\vert p$ and $n\vert a$ is $n=1$.

(2 Points )