Cubes and NinePro Problems > Math > Logic > Proofs
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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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:
Prove that there is no two digit number which is equal to the product of its digits.
Four consecutive integers are multiplied together. Determine, with proof, which digits could be in the ones place of the resulting product.
Prove that all perfect squares are either a multiple of 4, or one more than a multiple of 4.