What is the greatest common divisor of $11+i$ and $1+3i$? Or in general, how can we solve problems like this?
2026-04-04 05:42:20.1775281340
What is the greatest common divisor of $11+i$ and $1+3i$?
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You can try and guess the factorization of $1+3i$; since $$ (1+3i)(1-3i)=1+9=10=2\cdot 5 $$ the only candidates for being a prime factor are $1+i$, $2+i$ and $2-i$. Note that $1+i$ certainly divides both $1+3i$ and $1-3i$ (its conjugate is an associate).
Similarly, $$ (11+i)(11-i)=122=2\cdot 61 $$ so the only candidates are $1+i$, $6+5i$ and $6-5i$. The same argument as before applies.