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# Cubes and Nine

Pro 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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Problem by Mr. Twitchell

## Solution

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## Similar Problems

### Ones Place Proof

Four consecutive integers are multiplied together. Determine, with proof, which digits could be in the ones place of the resulting product.

### Squares Mod Four

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

### Product of Digits

Prove that there is no two digit number which is equal to the product of its digits.

### Rational Sine and Cosine

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
3
5
4
5
5
13
12
13
7
25
24
25
9
41
40
41

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