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I am almost certain of the answer to this question, but cannot seem to find something to confirm it, and I feel the proof is too simple to be correct. I have studied modular arithmetic a decent bit, but am too uncomfortable in the subject to feel confident in my answer. If this is a duplicate I apologize and request to be directed to the correct answer.

The question: If a number is congruent to 1 mod 4, can it ever be divisible by a number congruent to 3 mod 4. More specifically, for positive integers $a$ and $b$, we know $a \equiv 1 (\textrm{mod} 4)$ and $b \equiv 3 (\textrm{mod} 4)$. Can $b | a$?

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    $\begingroup$ 3|9$ $ $ $ $ $ $ $ $\endgroup$ – Exodd Apr 8 '17 at 17:12
  • $\begingroup$ Right--take any number that's 3 mod 4 and square it. $\endgroup$ – Elizabeth S. Q. Goodman Apr 8 '17 at 17:14
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Counterexample, for odd $n$: $$a=3^{2n}, b = 3^{n}$$

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  • $\begingroup$ Thanks, that was extremely simple. Must just be a bit brain-dead right now. Thanks for the help. $\endgroup$ – Nathan Apr 8 '17 at 17:16
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The product of two numbers congruent to $3$ mod $4$ is always congruent to $1$ mod $4$. For example, $3\cdot 7=21$.

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Here is my favourite example: Take $a=57$, the Grothendieck's Prime!!

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