Puzzles, Issue 05

Solve the puzzles that appeared in Issue 05.

These puzzles appeared in Issue 05 of the magazine.

Largest odd factors

Source: Daniel Griller
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?

Square factorials

Source: Tom Button (via Woody Lewenstein at Maths Jam)
Multiply together the first 100 factorials: $1!\times2!\times3!\times…\times100!$ Find a number, $n$, such that dividing this product by $n!$ produces a square number.

The answers to these puzzles are available here.

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