1984 AHSME Problems/Problem 5

Problem 5

The largest integer $n$ for which $n^{200}<5^{300}$ is

$\mathrm{(A) \ }8 \qquad \mathrm{(B) \ }9 \qquad \mathrm{(C) \ } 10 \qquad \mathrm{(D) \ }11 \qquad \mathrm{(E) \ } 12$

Solution

Since both sides are positive, we can take the $100th$ root of both sides to find the largest integer $n$ such that $n^2<5^3$. Fortunately, this is simple to evaluate: $5^3=125$, and the largest square less than $125$ is $11^2=121$, so the largest $n$ is $11, \boxed{\text{D}}$.

See Also

1984 AHSME (ProblemsAnswer KeyResources)
Preceded by
Problem 4
Followed by
Problem 6
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