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A potter makes more than $100$ but less than $300$ pots and arranges them in rows, with each row consisting of the same number of pots, for drying. He finds that if he places $6$ pots more per row, he can arrange the pots in $10$ less rows. How many pots does the potter make?

The answer to this question is given as $225$ but I obtained $120$ also as one of the possible solutions.

Is $120$ wrong? If yes, why?

Here's how I obtained the solutions:

Let the number of rows be $n$ and the number of pots per row be $x$ the we have $$nx=(n-10)(x+6)$$ Which is same as $$3n-5x=30$$ Solving it as a Diophantine equation, we obtain solutions for $(n,x)$ as $(10,0), (15,3), (20,6), (25,9), (30,12), \dots$

Since $100<nx<300$ so, we have solutions as $(20,6)$ and $(25,9)$.

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    $\begingroup$ You ought to explain your work, but I agree that $(120,20)$ is a valid solution, as is $(225,25)$. $\endgroup$
    – lulu
    Jun 8, 2022 at 11:04
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    $\begingroup$ @MathLover I'm editing my question, my apologies $\endgroup$
    – InanimateBeing
    Jun 8, 2022 at 11:35
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    $\begingroup$ Yes your work is correct. +1 $\endgroup$
    – Math Lover
    Jun 8, 2022 at 11:46

1 Answer 1

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My question was answered satisfactorily by (and thanks to) lulu and MathLover.
Since the question seemed unsolved on the outlook and a lot of time's passed so just mentioning the comments here:

Yes, the working is correct and $(120,20)$ is a valid solution, as is $(225,25)$.

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