# Proving: If n Divisible by 3, Then n^3 Divisible by 3

• sapiental
In summary, if the integer n is divisible by 3, then n^3 is also divisible by 3, as shown by the direct proof method.
sapiental

## Homework Statement

Prove the following: If the integer n is divisible by 3 then n^3 is divisible by 3.

Direct Proof

## The Attempt at a Solution

n = 3m

n^2 = 9m^2

n^2 = 3(3m^2)

I think the proof is done at this point because the 3 factors out but I also did this:

n^2 = 9m^2

(n^2)/9 = m^2

(n/3)(n/3) = (m)(m)

which also implies n is divisible by 3 since integer x integer = integer

My professor is kind of harsh on proofs so I am not sure if there are intermediate steps I'm missing. Thanks!

If 3 is a factor of n, it must also be a factor of any natural power of n. Do you mean n^3 or n^2? Your working looks like n^2. Anyway, the first way you did it is correct and satisfactory though if your teacher is really harsh, you may want to go like:

$$n= 3m$$ for some natural value of m, because n is divisible by 3 (Data).
Squaring both sides, $$n^2 = 9m^2 = 3 ( 3m^2)$$. Since m is natural, 3m^2 must also be natural, and hence 3 is a factor of n^2 as well.

## 1. What does it mean for a number to be divisible by 3?

When a number is divisible by 3, it means that it can be divided evenly by 3 without any remainder. In other words, the quotient when the number is divided by 3 is a whole number.

## 2. How do you prove that if n is divisible by 3, then n^3 is divisible by 3?

To prove this statement, we can use the definition of divisibility. If n is divisible by 3, then it can be written as n = 3k, where k is some integer. Substituting this into n^3, we get (3k)^3 = 27k^3, which is divisible by 3 since 27 is divisible by 3.

## 3. Can this statement be proven using mathematical induction?

Yes, this statement can be proven using mathematical induction. The base case is when n = 3, which is clearly divisible by 3. For the inductive step, we assume that the statement holds for some arbitrary value of n, and then show that it also holds for n+1. Thus, if n is divisible by 3, then (n+1)^3 = n^3 + 3n^2 + 3n + 1 is also divisible by 3.

## 4. Can you give an example to illustrate this statement?

For example, let n = 9. We can see that 9 is divisible by 3 since 9/3 = 3. Using the proof by substitution, we get (3)^3 = 27, which is clearly divisible by 3. Therefore, the statement holds for n = 9.

## 5. Why is this statement important in mathematics?

This statement is important in mathematics because it is a fundamental concept in number theory and algebra. It helps us to understand the relationship between divisibility and powers of numbers, and can be applied in various mathematical proofs and problem-solving strategies.

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