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2017 AIME I Problems/Problem 14

Revision as of 13:29, 3 January 2024 by Ddk001 (talk | contribs) (→‎Solution)

Problem 14

Let $a > 1$ and $x > 1$ satisfy $\log_a(\log_a(\log_a 2) + \log_a 24 - 128) = 128$ and $\log_a(\log_a x) = 256$. Find the remainder when $x$ is divided by $1000$.

Solution 1

The first condition implies

\[a^{128} = \log_a\log_a 2 + \log_a 24 - 128\]

\[128+a^{128} = \log_a\log_a 2^{24}\]

\[a^{a^{128}a^{a^{128}}} = 2^{24}\]

\[\left(a^{a^{128}}\right)^{\left(a^{a^{128}}\right)} = 2^{24} = 8^8\]

So $a^{a^{128}} = 8$.

Putting each side to the power of $128$:

\[\left(a^{128}\right)^{\left(a^{128}\right)} = 8^{128} = 64^{64},\]

so $a^{128} = 64 \implies a = 2^{\frac{3}{64}}$. Specifically,

\[\log_a(x) = \frac{\log_2(x)}{\log_2(a)} = \frac{64}{3}\log_2(x)\]

so we have that

\[256 = \log_a(\log_a(x)) = \frac{64}{3}\log_2\left(\frac{64}{3}\log_2(x)\right)\]

\[12 = \log_2\left(\frac{64}{3}\log_2(x)\right)\]

\[2^{12} = \frac{64}{3}\log_2(x)\]

\[192 = \log_2(x)\]

\[x = 2^{192}\]

We only wish to find $x\bmod 1000$. To do this, we note that $x\equiv 0\bmod 8$ and now, by the Chinese Remainder Theorem, wish only to find $x\bmod 125$. By Euler's Theorem:

\[2^{\phi(125)} = 2^{100} \equiv 1\bmod 125\]

so

\[2^{192} \equiv \frac{1}{2^8} \equiv \frac{1}{256} \equiv \frac{1}{6} \bmod 125\]

so we only need to find the inverse of $6 \bmod 125$. It is easy to realize that $6\cdot 21 = 126 \equiv 1\bmod 125$, so

\[x\equiv 21\bmod 125, x\equiv 0\bmod 8.\]

Using Chinese Remainder Theorem, we get that $x\equiv \boxed{896}\bmod 1000$, finishing the solution.

Solution 2 (Another way to find $a$)

$\log_a(\log_a(\log_a 2) + \log_a 24 - 128) = 128$

$\implies \log_a(\log_a 2))+log_a(24)=a^{128}+128$

$\implies \log_a(\log_a 2^{24})=a^{128}+128$

$\implies 2^{24}=a^{a^{(a^{128}+128)}}$

Obviously letting $a=2^y$ will simplify a lot and to make the $a^{128}$ term simpler, let $a=2^{\frac{y}{128}}$. Then,

$2^{24}=2^{\frac{y}{128} \cdot 2^{\frac{y}{128} \cdot (2^y+128)}}=2^{\frac{y}{128} \cdot 2^{y \cdot (2^{y-7}+1)}}$

$\implies 24=\frac{y}{128} \cdot 2^{y \cdot (2^{y-7}+1)}$

$\implies 3 \cdot 2^{10}=y \cdot 2^{y \cdot (2^{y-7}+1)}$

Obviously, y is $3$ times a power of $2$. (It just makes sense.) Testing, we see $y=6$ satisfy the equation so $a=2^{\frac{3}{64}}$. Therefore, $x=2^{192} \equiv \boxed{896} \pmod{1000}$

Alternate solution

If you've found $x$ but you don't know that much number theory.

Note $192 = 3 * 2^6$, so what we can do is take $2^3$ and keep squaring it (mod 1000).

\[2^3 = 8\] \[2^6 = 8*8 = 64\] \[2^{12} = 64*64 \equiv 96\bmod 1000\] \[2^{24} \equiv 96*96 \equiv 216\bmod 1000\] \[2^{48} \equiv 216*216 \equiv 656\bmod 1000\] \[2^{96} \equiv 656*656 \equiv 336\bmod 1000\] \[2^{192} \equiv 336*336 \equiv \boxed{896}\bmod 1000\]

Video Solution by mop 2024

https://youtu.be/E-7YQ9ND5Ms

~r00tsOfUnity

See also

2017 AIME I (Problems • Answer Key • Resources)
Preceded by
Problem 13 		Followed by
Problem 15
1 • 2 • 3 • 4 • 5 • 6 • 7 • 8 • 9 • 10 • 11 • 12 • 13 • 14 • 15
All AIME Problems and Solutions

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