2024 AMC 10A Problems/Problem 10: Difference between revisions
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== Solution 1 == | == Solution 1 == | ||
Let <math>s</math> be the number of times the operation is performed. Notice the sequence goes <math>100 \rightarrow 110 \rightarrow 120 \rightarrow 40 \rightarrow 50 \rightarrow 60 \rightarrow 20 \rightarrow 30 \rightarrow 10 \rightarrow 20 \rightarrow \cdots</math>. Thus, for <math>s \equiv 1 \pmod{3}</math>, the value is <math>30</math>. Since <math>100 \equiv 1 \pmod{3}</math>, the answer is <math>\boxed{\textbf{( | Let <math>s</math> be the number of times the operation is performed. Notice the sequence goes <math>100 \rightarrow 110 \rightarrow 120 \rightarrow 40 \rightarrow 50 \rightarrow 60 \rightarrow 20 \rightarrow 30 \rightarrow 10 \rightarrow 20 \rightarrow \cdots</math>. Thus, for <math>s \equiv 1 \pmod{3}</math>, the value is <math>30</math>. Since <math>100 \equiv 1 \pmod{3}</math>, the answer is <math>\boxed{\textbf{(C)} 30}</math> | ||
~andliu766 | ~andliu766 | ||
Revision as of 16:48, 8 November 2024
Problem
Consider the following operation. Given a positive integer 
, if is a multiple of 
, then you replace by 
. If is not a multiple of , then you replace by 
. Then continue this process. For example, beginning with 
, this procedure gives 
. Suppose you start with 
. What value results if you perform this operation exactly 
times?
Solution 1
Let 
be the number of times the operation is performed. Notice the sequence goes 
. Thus, for 
, the value is 
. Since 
, the answer is 
~andliu766
