2024 AMC 10B Problem 16
Problem 16 of 25IntermediateAlgebraNumber Theory
Jerry likes to play with numbers. One day, he wrote all the integers from to on the whiteboard. Then he repeatedly chose four numbers on the whiteboard, erased them, and replaced them by either their sum or their product. (For example, Jerry’s first step might have been to erase and and then write either their sum, or their product, on the whiteboard.) After repeatedly performing this operation, Jerry noticed that all the remaining numbers on the whiteboard were odd. What is the maximum possible number of integers on the whiteboard at that time?
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Solution
Among there are even numbers and odd numbers. Each operation replaces entries by so the total count falls by If a move consumes even entries, its output is either odd, reducing the even count by or even, reducing it by In either case the even count falls by at most Therefore eliminating all even entries takes at least moves. This is achievable: use sums containing one odd and three evens, then one sum containing three odds and the final even. Every output is odd. Thus the maximum remaining count is Therefore, the answer is A.