Difference between revisions of "2020 AMC 12A Problems/Problem 13"

Latest revision as of 17:48, 12 September 2023

Problem

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 $\boxed{\textbf{(B) } 3.}$

~Silverdragon

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.

~~BJHHar

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 $b=3$ .

~jackshi2006

@@ Line 41: / Line 41: @@
 ~~BJHHar
+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==
+<math>\sqrt[a]{N\sqrt[b]{N\sqrt[c]{N}}} = \sqrt[36]{N^{25}}</math> can be rewritten as <math>\frac{\frac{\frac{1}{c}+1}{b}+1}{a} = \frac{25}{36}</math>. The trick is to focus on the properties of the denominators and numerators. <math>a</math> must be either be a factor of 36, because in any other situation the denominator of the final expression is preserved, and <math>b</math> is forced to be 36, which is not an answer choice. Next, realize that the numerator of <math>\frac{\frac{1}{c}+1}{b}+1</math> must be <math>25</math>, and even more importantly, that the denominator must be large enough so that when <math>1</math> is subtracted from it, it will form a fraction less than <math>1</math>. The only number large enough to do this will also being a fraction of <math>36</math> is 18. This means that <math>a</math> is 2. Moving onward, we are left with <math>\frac{\frac{1}{c}+1}{b} = \frac{7}{18}</math>. Since <math>b</math> must be a factor of <math>18</math>, just plug in values from the answer choices to find that <math>c=6</math> and <math>b=3</math>.
+~jackshi2006
 ==See Also==

2020 AMC 12A (Problems • Answer Key • Resources)
Preceded by Problem 12	Followed by Problem 14
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All AMC 12 Problems and Solutions

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