# Eventual Pairs

Logic Level 5

$(10,4) \rightarrow (20,5) \rightarrow (21,10) \rightarrow (42,11) \rightarrow (43,22) \rightarrow (44,44)$

I have two integers $$A$$ and $$B$$. Each turn, I must double one number and add 1 to the other. I repeat this process, with the goal of making the integers equal to each other. The above is an example of how we can start from $$(10,4)$$ and get to two equal integers.

Given that I have the initial pairs of integers $$A = 300,B=301$$ and that after $$M$$ steps I have the pairs $$(N,N)$$ for some integer $$N$$. What is the value of $$N$$ such that $$M$$ is minimized?

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