Track / Stage 4 / 238 of 340 #498 of 1964

Problem 498

AMC 12 late, AIME early

Number theory Difficulty 4.9 Find the answer

1. (7 points) Compare $3^{76}$ and $5^{50}$ .

Your answer

A number or a short expression. Fractions can be typed as 3/2, and spacing doesn't matter.

Official solution

Solution. $3^{76}=3^{4} \cdot 3^{72}=81 \cdot\left(3^{3}\right)^{24}=81 \cdot(27)^{24}$ .

$5^{50}=5^{2} \cdot 5^{48}=25 \cdot\left(5^{2}\right)^{24}=25 \cdot(25)^{24}$ .

Since $81>25$ and $27>25$ , then $3^{76}>5^{50}$ .

Answer. $3^{76}>5^{50}$ .

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