2020 AMC 12A 第 13 题

Write the left side as $N$ to the power $\dfrac{1}{a} + \dfrac{1}{ab} + \dfrac{1}{abc}$

Set that exponent equal to $\dfrac{25}{36}$ and peel off one nested variable at a time starting with $a = 2$

The left side equals $N$ raised to the exponent $\dfrac{1}{a} + \dfrac{1}{ab} + \dfrac{1}{abc},$ which must equal $\dfrac{25}{36}.$

Because $b,c\ge2,$ this exponent is at most $\dfrac{7}{4a}.$ If $a\ge3,$ it is at most $\dfrac{7}{12}\lt\dfrac{25}{36},$ so $a=2.$ The equation becomes $\dfrac{1}{2b}+\dfrac{1}{2bc}=\dfrac{7}{36}.$

Clearing denominators gives $c(7b-18)=18.$ Set $d=7b-18,$ so $d$ is a positive divisor of $18.$ Among $1,2,3,6,9,18,$ only $d=3$ makes $b=\dfrac{d+18}{7}$ an integer. Hence $b=3$ and $c=6.$

Thus, B is the correct answer.