Determine 8611

Determine all natural numbers A and B pairs for which the sum of twice the least common multiple and three times the greatest common divisor of natural numbers A and B is equal to their product.

Correct answer:

a₁ =  5
b₁ =  5
a₂ =  9
b₂ =  3

Step-by-step explanation:

$$\begin{gathered} 2LCM(a,b)+3GCD(a,b)=ab \\ {a}_1=5 \\ {b}_1=5 \\ 5...\text{ prime number} \\ \text{GCD}(5,5)=5 \\ 5...\text{ prime number} \\ LCM(5,5)=5 \\ {l}_1=2\cdot LCM(a_1,b_1)+3\cdot GCD(a_1,b_1)=2\cdot LCM(5,5)+3\cdot GCD(5,5)=25 \\ {r}_1=5\cdot 5=25 \end{gathered}$$

$b_1=5$

$$\begin{gathered} 2LCM(a,b)+3GCD(a,b)=ab \\ {a}_2=9 \\ {b}_2=3 \\ 9=3^2 \\ 3...\text{ prime number} \\ \text{GCD}(9,3)=3 \\ 9=3^2 \\ 3...\text{ prime number} \\ LCM(9,3)=3^2=9 \\ {l}_2=2\cdot LCM(a_2,b_2)+3\cdot GCD(a_2,b_2)=2\cdot LCM(9,3)+3\cdot GCD(9,3)=27 \\ {r}_2=9\cdot 3=27 \end{gathered}$$

$b_2=3$

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