Competition problem — Number theory

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Number theory Difficulty 1.9 Junior Find the answer

What is the correct ordering of the three numbers, $10^8$ , $5^{12}$ , and $2^{24}$ ?
$\textbf{(A)}\ 2^{24}<10^8<5^{12}\\ \textbf{(B)}\ 2^{24}<5^{12}<10^8 \\ \textbf{(C)}\ 5^{12}<2^{24}<10^8 \\ \textbf{(D)}\ 10^8<5^{12}<2^{24} \\ \textbf{(E)}\ 10^8<2^{24}<5^{12}$

Your answer

Multiple choice: answer with the letter of the option you want.

Solution

Use brute force.
$10^8=100,000,000$ ,
$5^{12}=244,140,625$ , and
$2^{24}=16,777,216$ .
Therefore, $\boxed{\text{(A)}2^{24}<10^8<5^{12}}$ is the answer. (Not recommended for the contest and will take forever)

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Source: NuminaMath-1.5, licensed Apache-2.0. Statement and solution reproduced as published; topic and difficulty added by this site.