MHB Elementary Number Theory proof

[cbarker1]

Dear Everyone,

Here is the question:

"Prove that if $k$ divides the integers $a$ and $b$, then $k$ divides $as+bt$ for every pair of integers $s$ and $t$ for every pair of integers."

The attempted work:

Suppose $k$ divides $a$ and $k$ divides $b$, where $a,b\in\mathbb{Z}$. Then, $a=kt$ and $b=ks$, where $s,t\in\mathbb{Z}$ (Here is where I am stuck).

Do I solve for $k$?

If I do solve for $k$, then it yields the
$k=\frac{a}{t}$. Then, $b=\frac{as}{t}$. So $bt-as=0$. Then $0$ divides $bt-as$.

So is it right to the proof this way?

 

[castor28]

Dear Everyone,

Here is the question:

"Prove that if $k$ divides the integers $a$ and $b$, then $k$ divides $as+bt$ for every pair of integers $s$ and $t$ for every pair of integers."

The attempted work:

Suppose $k$ divides $a$ and $k$ divides $b$, where $a,b\in\mathbb{Z}$. Then, $a=kt$ and $b=ks$, where $s,t\in\mathbb{Z}$ (Here is where I am stuck).

Do I solve for $k$?

If I do solve for $k$, then it yields the
$k=\frac{a}{t}$. Then, $b=\frac{as}{t}$. So $bt-as=0$. Then $0$ divides $bt-as$.

So is it right to the proof this way?

Hi Cbarker1,

$s$ and $t$ are arbitrary integers in the question, you should not use them as you do in the proof, where they are fixed integers that depend on $a$, $b$, and $k$.

You can say that there are integers $m$ and $n$ such that $a=km$ and $b=kn$. Now, you have:
$$
as+bt = (km)s + (kn)t = k(ms+nt)
$$
As $ms+nt$ is an integer, this shows that $k$ divides $as+bt$.