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Find the greatest number that will divide $2327, 2677, 4007$ and $497$ and will leave a remainder of $17, 37, 47$ and $57$ respectively.

My attempt:

Since all these numbers leave a remainder,

$\begin{align}2327 -17 &\to 2310 \tag1\\ 2677 -37 &\to 2640 \tag2 \\ 4007 -47 &\to 3960 \tag3\\ 497 -57 &\to 440 \tag4\end{align}$

Now simply by brute force method (took me a couple of tries) I ended up with the result $110$.

$\begin{align} 2640-2310&=330 && \tag 5 \\ 440-330 & = 110 \tag6\end{align} $

This result seems to work for me. I checked the remainders after subtracting and they all comply.

But the question explicitly asks for the the greatest result. How do I know there are no better answers?

And also is there a faster way to solve it? (This was a MCQ and we are supposedly to spend less than 2 min per question and I took well over 10).

Thanks!

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    The problem statement is confusing: A number that will divide 2327 etc. won't leave a remainder. – 2017-01-30
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    @HagenvonEitzen It's an interview quantitative test question. I've reproduced it verbatim. I guess the question setter isn't all that keen on grammar! $\ddot \smile$ – 2017-01-30
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    You know there is no greater answer because the value must divide both 440 and 330. And you know there is no *lesser* answer because it must be greater than 57. – 2017-01-30
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    @Joffan Yes you're right... Now to think of it, answer was right there all along! – 2017-01-30

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You are correct in your method. $$2327 \equiv 17 \pmod {x} \iff 2310 \equiv 0\pmod x$$ $$2677 \equiv 37 \pmod {x} \iff 2640 \equiv 0\pmod x$$ $$4007 \equiv 47 \pmod {x} \iff 3960 \equiv 0\pmod x$$ $$497 \equiv 57 \pmod {x} \iff 440 \equiv 0\pmod x$$ Thus $$x=\gcd(3960, 2640, 2310, 440)=\gcd((\gcd(3960, 2640), \gcd(2310, 440))$$ Now apply the Euclidean Algorithm, to get, as you got, $110$. So your answer is correct.

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    The *Euclidean Algorithm*. Right. Thanks! – 2017-01-30
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    @Macindows Thanks. – 2017-01-30
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    @Macindows The arrows should go in *both* directions else the proof is not complete. – 2017-01-30
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    @BillDubuque Technically so, but I think the OP can understand it even if that is so. And I cannot edit it right now – 2017-01-31
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    Yes I did understand the result no issues. @BillDubuque In any case I submitted the edited version of the answer. Kudos. – 2017-01-31