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I have a question regarding a division algorithm proof. I know the question has been posted here but I am confused with a very specific step. Here is the question:

$$\gcd(a,4)=2$$ $$\gcd(b,4)=2$$

so find the $\gcd(a+b,4)$ and prove it. Here is what I have so far:

I know that since $2|a$ then and $2|b$ then $a$ and $b$ must be even. So $a=2k$ and $b=2j$ for $k$, $j$ are integers. Then from here I know I need to use the division algorithm but how what is my divisor? do I need to work backwards? Thanks in advance

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    $\gcd(a,4)=2$ tells you more than just $2 \mid a$, it tells you that no higher power of $2$ divides $a$. Therefore the $k,j$ you determined must be odd, then the conclusion follows easily. – 2017-02-05
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    Okay, so this means: 2|a so a=2k and k must be odd k=2q+1 , 2|b so b=2j and j must be odd so j=2h+1 adding these equations together we get 4p+4h+4 so we can factor 4 and thats the answer? i just want to make sure I get it. – 2017-02-05
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    Right, that's it. FWIW it's the same as Ethan Bolker's hint. – 2017-02-05
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    Thank you guys so much@Ethan @dxiv. It seems like there is no need for division algorithm after all. – 2017-02-05

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Hint. I don't think you need the division algorithm, you just need to know something about divisibility.

Your hypotheses tell you that $2$ divides $a$ but $4$ does not, so $a$ is twice an odd number. The same is true for $b$. What can you conclude about $a+b$?

Edit. Your comments show that you know how to solve the problem now, without the division algorithm. When you learn about congruences you'll do it this way: $$ a \equiv 2 \pmod{4} \quad\text{and} \quad b \equiv 2 \pmod{4} $$ imply $$ a + b \equiv 2 + 2 \equiv 0 \pmod{4} $$ so $\gcd(a+b,4) = 4$.

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    Thank you so much. I am not sure why my professor suggested i use D.A. I think that if a and b are even then we can factor out a 2 out of them? – 2017-02-05
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    Yes, but you can't take out two $2$s. – 2017-02-05
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    is there a way to do it using D.A? – 2017-02-05
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    @HananIbraheim I don't think so. You would start by writing $a+b = 4q +r$ and figuring out the remainder $r$. To do that you would have to solve the problem first. – 2017-02-05
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    how did you write a+b=4q+1 like why? – 2017-02-05
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    $a+b$ is not equal to $4q+1$. It's equal to $4q+r$ and you have to figure out $r$. If you follow my hint you should be able to show why $r$ must be $0$. – 2017-02-05
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    Can you explain how you figured that a+b=4q+r? If I let a and b be both even then they are equal to say: a=2k , b =2j basically can we combine them the only way we could do that if they are equal but how do we prove that? – 2017-02-05