Adela number

Adela had two numbers written on the paper. When she added their greatest common divisor and least common multiple, she was given four different numbers less than 100. She was amazed that if she divided the largest of these four numbers by the least, she would get the largest
common divisor of all four numbers. What numbers did Adela have written on the paper?

Final Answer:

a =  12
b =  18

Step-by-step explanation:

$a=12$

$$\begin{gathered} b=18 \\ 12=2^2\cdot 3 \\ 18=2\cdot 3^2 \\ LCM(12,18)=2^2\cdot 3^2=36 \\ n=LCM(a,b)=LCM(12,18)=36 \\ 12=2^2\cdot 3 \\ 18=2\cdot 3^2 \\ \text{GCD}(12,18)=2\cdot 3=6 \\ m=GCD(a,b)=GCD(12,18)=36 \\ a<100,b<100,n<100,m<100 \\ 6;12;18;36 \\ nsd(6,12,18,36)=6 \\ 36/6=6 \end{gathered}$$

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