Pole cutting pieces

Mr. Novák needs three wooden poles 21 dm, 350 cm, and 2.8 m long. He needs to cut them into smaller pieces of the same length, as long as possible. How long will he get and how many pieces will he have altogether?

Final Answer:

x =  70 cm
n =  12

Step-by-step explanation:

$$\begin{gathered} a=21\text{dm}\to \text{cm}=21\cdot 10\text{cm}=210\text{cm} \\ b=350\text{cm} \\ c=2.8\text{m}\to \text{cm}=2.8\cdot 100\text{cm}=280\text{cm} \\ 210=2\cdot 3\cdot 5\cdot 7 \\ 350=2\cdot 5^2\cdot 7 \\ 280=2^3\cdot 5\cdot 7 \\ \text{GCD}(210,350,280)=2\cdot 5\cdot 7=70 \\ x=GCD(a,b,c)=GCD(210,350,280)=70\text{cm} \end{gathered}$$

$$\begin{gathered} d=a+b+c=210+350+280=840\text{cm} \\ n=d/x=840/70=12 \end{gathered}$$

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