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Largest odd factors

Pick a number. Call it \(n\). Write down all the numbers from \(n+1\) to \(2n\) (inclusive). For example, if you picked 7, you would write:
$$8,9,10,11,12,13,14$$
Below each number, write down its largest odd factor. Add these factors up. What is the result? Why?

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If you enjoyed this puzzle, check out Sunday Afternoon Maths LVII,
puzzles about odd numbers, or a random puzzle.
© Matthew Scroggs 2019