2020 AMC 12A Problems/Problem 13

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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 $N \neq 1$. What is $b$?

$\textbf{(A) } 2 \qquad \textbf{(B) } 3 \qquad \textbf{(C) } 4 \qquad \textbf{(D) } 5 \qquad \textbf{(E) } 6$

Solution 1

$\sqrt[a]{N\sqrt[b]{N\sqrt[c]{N}}}$ can be simplified to $N^{\frac{1}{a}+\frac{1}{ab}+\frac{1}{abc}}.$

The equation is then $N^{\frac{1}{a}+\frac{1}{ab}+\frac{1}{abc}}=N^{\frac{25}{36}}$ which implies that $\frac{1}{a}+\frac{1}{ab}+\frac{1}{abc}=\frac{25}{36}.$

$a$ has to be $2$ since $\frac{25}{36}>\frac{7}{12}$. $\frac{7}{12}$ is the result when $a, b,$ and $c$ are $3, 2,$ and $2$

$b$ being $3$ will make the fraction $\frac{2}{3}$ which is close to $\frac{25}{36}$.

Finally, with $c$ being $6$, the fraction becomes $\frac{25}{36}$. In this case $a, b,$ and $c$ work, which means that $b$ must equal $\boxed{\textbf{(B) } 3.}$~lopkiloinm

Solution 2

As above, notice that you get $\frac{1}{a}+\frac{1}{ab}+\frac{1}{abc}=\frac{25}{36}.$

Now, combine the fractions to get $\frac{bc+c+1}{abc}=\frac{25}{36}$.

Let us assume $bc+c+1=25$ and $abc=36$ as this is the most convenient. (EDIT: This used to say WLOG but that is inaccurate)

From the first equation, we get $c(b+1)=24$. Note also that from the second equation, $b$ and $c$ must be factors of 36.

After listing out the factors of 36 and utilising trial and error, we find that $c=6$ and $b=3$ works, with $a=2$. So our answer is $\boxed{\textbf{(B) } 3.}$


Edits by ~Snore, ~Swaggergotcha

Solution 3

Collapsed, $\sqrt[a]{N\sqrt[b]{N\sqrt[c]{N}}} = \sqrt[abc]{N^{bc+c+1}}$. Comparing this to $\sqrt[36]{N^{25}}$, observe that $bc+c+1=25$ and $abc=36$. The first can be rewritten as $c(b+1)=24$. Then, $b+1$ has to factor into 24 while 1 less than that also must factor into 36. The prime factorizations are as follows $36=2^2 3^2$ and $24=2^33$. Then, $b=\boxed{\textbf{B)}3}$, as only 4 and 3 factor into 36 and 24 while being 1 apart.


Note: 3 and 2 also factor into 24 and 36 while being 1 apart, but we can eliminate by quick guess and check. ~ Edit by Soupnoodle

Solution 4

$\sqrt[a]{N\sqrt[b]{N\sqrt[c]{N}}} = \sqrt[36]{N^{25}}$ can be rewritten as $\frac{\frac{\frac{1}{c}+1}{b}+1}{a} = \frac{25}{36}$. The trick is to focus on the properties of the denominators and numerators. $a$ must be either be a factor of 36, because in any other situation the denominator of the final expression is preserved, and $b$ is forced to be 36, which is not an answer choice. Next, realize that the numerator of $\frac{\frac{1}{c}+1}{b}+1$ must be $25$, and even more importantly, that the denominator must be large enough so that when $1$ is subtracted from it, it will form a fraction less than $1$. The only number large enough to do this will also being a fraction of $36$ is 18. This means that $a$ is 2. Moving onward, we are left with $\frac{\frac{1}{c}+1}{b} = \frac{7}{18}$. Since $b$ must be a factor of $18$, just plug in values from the answer choices to find that $c=6$ and $b=3$.


See Also

2020 AMC 12A (ProblemsAnswer KeyResources)
Preceded by
Problem 12
Followed by
Problem 14
1 2 3 4 5 6 7 8 9 10 11 12 13 14 15 16 17 18 19 20 21 22 23 24 25
All AMC 12 Problems and Solutions

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