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# Puzzles

## 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?

## Odd squares

Source: Maths Jam
Prove that 1 and 9 are the only square numbers where all the digits are odd.

## Odd sums

What is $$\frac{1+3}{5+7}$$?
What is $$\frac{1+3+5}{7+9+11}$$?
What is $$\frac{1+3+5+7}{9+11+13+15}$$?
What is $$\frac{1+3+5+7+9}{11+13+15+17+19}$$?
What is $$\frac{\mathrm{sum\ of\ the\ first\ }n\mathrm{\ odd\ numbers}}{\mathrm{sum\ of\ the\ next\ }n\mathrm{\ odd\ numbers}}$$?

## Archive

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