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Number theory Difficulty 1.4 Junior Find the answer Canada

An integer $m$ is chosen at random from the list $-9,-7,-5,-3,-1,1,3,5,7,9$ . The probability that $m^4>100$ is

Pick one

Solution

There are 10 numbers in the list.

We note that $(-1)^4 = 1^4 = 1 \qquad (-3)^4 = 3^4 = 81 \qquad (-5)^4 = 5^4 = 625$ $(-7)^4 = 7^4 = 2401 \qquad (-9)^4 = 9^4 = 6561$ Thus, if $m=-3,-1,1,3$ , then $m^4<100$ . If $m=-9,-7,-5,5,7,9$ , then $m^4>100$ .

In other words, there are exactly six numbers in the list whose fourth power is larger than 100.

Thus, if $m$ is chosen at random from the list, the probability that $m^4>100$ is $\frac{6}{10}$ or $\frac{3}{5}$ .

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Source: CEMC, University of Waterloo, licensed CC-BY-NC-4.0. Statement reproduced verbatim; metadata (topic, difficulty) added by this project.