Competition problem — Number theory

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Number theory Difficulty 2.2 Junior Find the answer

$\sqrt{164}$ is
$\text{(A)}\ 42$ $\text{(B)}\ \text{less than }10$ $\text{(C)}\ \text{between }10\text{ and }11$ $\text{(D)}\ \text{between }11\text{ and }12$ $\text{(E)}\ \text{between }12\text{ and }13$

Your answer

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

Solution

Note that if $1\leq a<b<c$ , then $\sqrt{a}<\sqrt{b}<\sqrt{c}$ .
Since $1<144<164<169$ , we can say that $\sqrt{144}<\sqrt{164}<\sqrt{169} \Rightarrow 12<\sqrt{164}<13 \rightarrow \boxed{\text{E}}$

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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.