There are integers a, b, and c, each greater than 1, such that √[a]N √[b]N √[c]N = √[36]N^25 for all N \gt 1. What is b?

2020 AMC 12A Problem 13 — Solution

2020 AMC 12A Problem 13

Below is the professionally curated solution for Problem 13 of the 2020 AMC 12A, from LIVE by Po-Shen Loh. You can also try the full timed exam, view all 2020 AMC 12A solutions, or check the answer key.

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Concepts:radical exponent

Difficulty rating: 1590

13.

There are integers $a,$ $b,$ and $c,$ each greater than $1,$ such that $\sqrt[a]{N \sqrt[b]{N \sqrt[c]{N}}} = \sqrt[36]{N^{25}}$ for all $N \gt 1.$ What is $b?$

$2$

$3$

$4$

$5$

$6$

Solution:

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

Trying $a = 2$ leaves $\dfrac{1}{2b} + \dfrac{1}{2bc} = \dfrac{25}{36} - \dfrac12 = \dfrac{7}{36}.$

Then $\dfrac{1}{2b}\left(1 + \dfrac1c\right) = \dfrac{7}{36}.$ Taking $b = 3$ gives $1 + \dfrac1c = \dfrac{7}{6},$ so $c = 6.$

All three are integers greater than $1,$ and $b = 3.$

Thus, B is the correct answer.

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