Is the following a valid proof in number theory?

Question

Prove the following using fundamental definitions.

Suppose $a$, $b$, and $c$ are positive integers such that $a|b$ and $a|c$. Then for any integer $k$, $ka + 3b ≡ c \mod a)$

What I've done:

"We need to show $ka + 3b ≡ c(mod a)$. So by the definition of congruent we need to show $ka + 3b - c = am$ with $m$ being an integer. So, we need to show $a|(ka + 3b -c)$.

Notice then that $a|(ka + 3b -c)$ since $a>0$

Is this a valid proof?

My professor marked my proof incorrect so i was wondering if something did not make sense that i overlooked — Maru, Apr 02 '20 at 17:27
@Maru, you can add that when $x|y$ and $x|z$, then $x|(y+z)$. also the word fundamental definitions. is a bit ambiguous, cuz it depends on your previous knowledge — Aven Desta, Apr 02 '20 at 17:29

Surb · Answer 1 · 2020-04-02T17:35:35.253

1

Notice that since $a\mid c$, $$c\equiv 0\pmod a.$$

Now, since $a\mid b$, then $$ka+3b\equiv 0\pmod a.$$

The claim follow.

edited Apr 02 '20 at 17:35

answered Apr 02 '20 at 17:18

Surb

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Is the following a valid proof in number theory?

1 Answers1