Different 5402

Adélka 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 Adelka have written on the paper?

Correct 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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