Is Chinese Remainder Theorem relevant here?

Number Theory Level 5

$\large{\begin{cases} Z &=& (q_{1})(1 - i) \\ Z &=& (q_{2})(1 - 2i) - i \\ Z &=& (q_{3})(1 - 4i) + 2 \\ Z &=& (q_{4})(3 - 2i) + 2i \\ \end{cases} }$

Let $Z$ be a Gaussian integer and $q_{1}$ , $q_{2}$ , $q_{3}$ , & $q_{4}$ be the Gaussian integral quotients of the system above, what is the least value of $| Z | ^{2}$ ?

The answer is 178.

1 solution

Mark Hennings
Jan 9, 2020

Since the Gaussian integers form a Euclidean domain, the Chinese Remainder Theorem is applicable, and we note that $1-i$ , $1-2i$ , $1-4i$ and $3-2i$ are coprime in the Gaussian integers. We obtain that $Z \; = \; (37 - 29i)p - 195 - 11i \hspace{2cm} p \in \mathbb{Z}[i]$ and it is easy to show that $|Z|^2$ is minimized when $p = 3 + 3i$ , giving the minimum value of $|Z|^2$ as $|3 + 13i|^2 = \boxed{178}$ .

Is Chinese Remainder Theorem relevant here?

The answer is 178.

1 solution

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