MO C–I–1 2018

An unknown number is divisible by just four numbers from the set {6, 15, 20, 21, 70}. Determine which ones.

Result

x = (Correct answer is: 6,15,21,70)

Step-by-step explanation:

$$\begin{gathered} 6=2\cdot 3 \\ 15=3\cdot 5 \\ 20=2^2\cdot 5 \\ 21=3\cdot 7 \\ 70=2\cdot 5\cdot 7 \\ LCM(6,15,20,21,70)=2^2\cdot 3\cdot 5\cdot 7=420 \\ a=LCM(6,15,20,21,70)=420 \\ {x}_1=210 \\ {x}_2=630 \\ {x}_3=1050 \\ {x}_4=1470 \\ {x}_5=1890 \\ {x}_6=2310 \\ {x}_7=2730 \\ \dots \\ d(210)=1,2,3,5,6,7,10,14,15,21,30,35,42,70,\dots \\ d(630)=1,2,3,5,6,7,9,10,14,15,18,21,30,35,42,45,63,70,\dots \\ LCM(6,15,21,70)=2\cdot 3\cdot 5\cdot 7=210 \\ k=LCM(6,15,21,70)=210 \\ x=6,15,21,70 \end{gathered}$$

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