Exercise 1.2.12 (If $(a,4) = 2$ and $(b,4) =2$, then $(a+b,4) = 4$. )

Question

Given that $(a,4) = 2$ and $(b,4) = 2$, prove that $(a+b,4) = 4$.

Richard Ganaye · Accepted Answer

ewcommand{\Z}{\mathbb{Z}}

\begin{proof} Assume that $(a,4) = 2$. Every $a \in \Z$ is of the form $4k+r,\ 0\leq r < 4$. Moreover $(4k,4) = 4, (4k+1,4) =  (1,4) = 1, (4k+3, 4) = (3,4) = 1$. This shows that 
$$a = 4k+2, \ k \in \Z.$$
Similarly, if $(b,4) = 2$, then $b = 4l +2$ for some $l \in \Z$.

Then $a+b = 4(k+l) + 4$, so $4 \mid a+b$. This proves $(a+b,4) = 4$.
\end{proof}

Exercise 1.2.12 (If $(a,4) = 2$ and $(b,4) =2$, then $(a+b,4) = 4$. )

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Exercise 1.2.12 (If $(a,4) = 2$ and $(b,4) =2$, then $(a+b,4) = 4$. )

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