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Problem 510

AMC 12 late, AIME early

Number theory Difficulty 4.9 Find the answer

$6 \cdot 74^{16} \cdot 5^{25}$ The number of digits (in decimal notation) of this number is
(A) 31 .
(B) 30 .
(C) 29 .
(D) 28 .
(E) 27 .
(35th American High School Mathematics Examination, 1984)

Your answer

Official solution

[Solution] Express $4^{16} 5^{25}$ in scientific notation (exponential notation), we have
$4^{16} \cdot 5^{25}=2^{32} \cdot 5^{25}=2^{7} \cdot 10^{25}=128 \cdot 10^{25} \text {. }$

This indicates that there are 25 zeros following 128, so there are a total of 28 digits. Therefore, the answer is $(D)$ .

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