Euler problem

Someone buys a number of towels for a total of 720 tolars. If for the same amount of money they could buy 6 more towels, each towel would be 6 tolars cheaper. How many towels were originally bought?

Final Answer:

n =  24

Step-by-step explanation:

$$\begin{gathered} 720=c\cdot n \\ 720=(c-6)\cdot (n+6);n>0 \\ 720=(720/n-6)\cdot (n+6) \\ 720\cdot n=(720-6\cdot n)\cdot (n+6) \\ 720\cdot n=(720-6\cdot n)\cdot (n+6) \\ 6n^2+36n-4320=0 \\ 6=2\cdot 3 \\ 36=2^2\cdot 3^2 \\ 4320=2^5\cdot 3^3\cdot 5 \\ \text{GCD}(6,36,4320)=2\cdot 3=6 \\ {n}^2+6n-720=0 \\ a=1;b=6;c=-720 \\ D=b^2-4ac=6^2-4\cdot 1\cdot (-720)=2916 \\ D>0 \\ {n}_{1,2}=\frac{-b\pm \sqrt{D}}{2a}=\frac{-6\pm \sqrt{2916}}{2} \\ {n}_{1,2}=\frac{-6\pm 54}{2} \\ {n}_{1,2}=-3\pm 27 \\ {n}_1=24 \\ {n}_2=-30 \\ n=n_1=24 \\ \text{ Verifying Solution: } \\ c=720/n=720/24=30 \\ {t}_1=c\cdot n=30\cdot 24=720 \\ {t}_2=(c-6)\cdot (n+6)=(30-6)\cdot (24+6)=720 \end{gathered}$$

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