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The Secretary of the School Puzzle Solvers Society sends the Treasurer to the stationery shop to buy as many ballpoint pens and notebooks as she can lay her hands on. The price of a pen is $e2.45$ and the price of a notebook is $e1.60$. On her return, the Treasurer says that she spent a total of $e419.45$. (a) Write out the above information using a linear Diophantine equation and explain the meaning of any symbols you use. (b) Find all possible solutions to the Diophantine equation that reveal how many pens and how many notebooks the Treasurer may have bought

I have so far got the equation $$245x+160y=41945$$

I have found the $$\gcd(245,160) = 5$$

I am now trying to roll back but that confuses me.

I let $5 = 75-7\cdot 10$

Am i on the right track?

Peter
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Sean3
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    What have you tried? – Dietrich Burde May 25 '21 at 15:10
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    Welcome to MSE. You'll get a lot more help if you show that you have made a real effort to solve the problem yourself, even if you haven't made much progress. What are your thoughts? What have you tried? How far could you get? Where are you stuck? This question will likely be closed if you don't add more context. Please respond by editing the question body. Clarifications don't belong in the comments. – saulspatz May 25 '21 at 15:13
  • You want to solve $245x+160y=41945$ for integer solutions, a Bézout relation might help with that – Evariste May 25 '21 at 15:14
  • Hi Sean. There are many similar problems solved on our site, for example the link I paste below. Please take a look at these and try to imitate the set-up of the equation for your pens and notebooks. https://math.stackexchange.com/questions/985374 – 311411 May 25 '21 at 15:15
  • I have just started this module and we have not done a word problem example so i am very confused .I have made it a linear Diophantine equation of 2.45x + 1.6y = 419.45. I then multipled that all by 20 to get rid of decimals. I then used euclids algorithim to find the gcd . which i think is 1 – Sean3 May 25 '21 at 15:16
  • @Sean Diophantine equations have integer coefficients, multiply both sides by $100$ (or better yet, $100/\gcd(245,160)=20$) – Evariste May 25 '21 at 15:18
  • I have the gcd to be 5. is that correct? – Sean3 May 25 '21 at 15:24
  • i used euclids algorithim – Sean3 May 25 '21 at 15:26
  • Ok. If you used the extended Euclid's algorithm, you should now have $x$ and $y$ that satisfy $245x+160y=5$. So you should [edit] that info into your question. BTW, there are 5 positive solution pairs to $245x+160y=41945$ – PM 2Ring May 26 '21 at 15:03

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