Smallest Number n

Algebra Level 2

What is the smallest value of the positive integer n n such that n 300 > 3 500 n^{300}>3^{500}

343 8 6 7 244

Achmad Damanhuri by Achmad Damanhuri , 26, Indonesia

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3 solutions

n 300 > 3 500 n > 3 5 3 n > 3 2 − 1 3 n > 9 3 1 3 n 3 > 729 3 n 3 > 243 ⟹ n ≥ 7 \begin{aligned}n^{300}&>3^{500}\\\\ n&>3^{\tfrac{5}{3}}\\\\ n&>3^{2-\tfrac{1}{3}}\\\\ n&>\dfrac{9}{3^{\tfrac{1}{3}}}\\\\ n^3&>\dfrac{729}{3}\\\\ n^3&>243\\\\ \implies n&\geq\color{#EC7300}\boxed{\color{#333333}7}\end{aligned}

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Mr. India
Apr 8, 2019

n 300 > 3 500 n^{300}>3^{500}

⇒ ( n 3 ) 100 > ( 3 5 ) 100 \Rightarrow (n^3)^{100}>(3^5)^{100}

⇒ n 3 > 3 5 → n 3 > 243 \Rightarrow n^3>3^5 \rightarrow n^3>243

⇒ n m i n = 7 \Rightarrow n_{min}=7

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n>3 to the power of (5/3) or 6.24025. Hence the smallest integral value of n is 7

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