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Cubes and Nine

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Cubes and Nine

Prove that all perfect cubes are either a multiple of 9, one more than a multiple of 9, or one less than a multiple of 9.

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Problem by Mr. Twitchell

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Prove that all perfect squares are either a multiple of 4, or one more than a multiple of 4.

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There are infinitely many angles θ between 0º and 90º for which both sin θ and cos θ are rational numbers. Prove this statement.

Hint: Here are some examples:

 sin θ
cos θ
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