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I was studying CRC from lecture notes of Schwarzkopf but I am stuck at this statement:

If $G(x)$ is a factor of $E(x)$, then $G(1)$ would also have to be $1$.

What does this statement means? It is the third point in the third last block of the article whose link is given. I know a similar question is asked but that is a specific question. I want to understand this particular statement.

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  • $\begingroup$ Suppose that $E(x) = G(x) H(x)$ and $E(x) = 1$. Then also $G(x) = 1$, since we're working modulo 2. $\endgroup$ Feb 17 '21 at 13:51
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Recall that we are working modulo $2$. Thus $E(x)$ is a polynomial whose coefficients are $0,1$, and $E(1) \in \{0,1\}$.

By definition, $G(x)$ is a factor of $E(x)$ if there exists a polynomial $H(x)$ such that $E(x) = G(x) H(x)$.

In this case, we assume that $E(1) = 1$. Since $E(1) = G(1) H(1)$, this forces $G(1) = 1$.

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