# 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.

Presentation mode

Problem by Mr. Twitchell

## Solution

In order to make it feasible for teachers to use these problems in their classwork, no solutions are publicly visible, so students cannot simply look up the answers. If you would like to view the solutions to these problems, you must have a Virtual Classroom subscription.Assign this problem

Click here to assign this problem to your students.## Similar Problems

### 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

### Product of Digits

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

### 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.

# Featured Games on This Site

Match color, font, and letter in this strategy game

Mastermind variation, with words

# Blogs on This Site

Reviews and book lists - books we love!

The site administrator fields questions from visitors.

Like us on Facebook to get updates about new resources