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Order 5 of the following sentences so that they form a logical proof of the statement:

a divides b and a divides c implies a divides b+c

Choose from this list of propositions

$\exists k, j$ such that $b = ka$ and $c = ja$
$\exists k$ such that $b = ka$ and $c = ka$
$b + c = ka + ja$
$b + c = (2k)a$
Assume for some integers $a, b$ and $c$ that $b | a$ and $c | a$ .
$a$ divides $b+c$
Assume $a | b$ and $a | c$ for some integers $a, b$ and $c$
$b + c = ka + ka$
$b + c = a ( k + j)$

Proof of the statement above (in order):

Choose from this list of propositions

$\exists k, j$ such that $b = ka$ and $c = ja$
$\exists k$ such that $b = ka$ and $c = ka$
$b + c = ka + ja$
$b + c = (2k)a$
Assume for some integers $a, b$ and $c$ that $b | a$ and $c | a$ .
$a$ divides $b+c$
Assume $a | b$ and $a | c$ for some integers $a, b$ and $c$
$b + c = ka + ka$
$b + c = a ( k + j)$

Proof of the statement above (in order):