Question

For which elements \(n \in\\{0,1,2,3,4,5\\}\) does 6 divide \(n^{3}+5 n ?\)

Short answer

The elements are \(n = 0, 2, 4\).

Step-by-step solution

Understand the Problem

We need to find elements \(n\) within the set \(\{0,1,2,3,4,5\}\) such that the expression \(n^3 + 5n\) is divisible by 6. This means that \(n^3 + 5n\) should equal \(6k\) for some integer \(k\).

Setup Expression for Divisibility

A number is divisible by 6 if it is divisible by both 2 and 3. Therefore, \(n^3 + 5n\) must be divisible by 2 and by 3. Let's test each element for these conditions.

Test Divisibility by 2

A number is divisible by 2 if it is even. The expression \(n^3 + 5n\) can be simplified to \(n(n^2 + 5)\). Note that \(n\) is multiplied by \((n^2 + 5)\). If \(n\) is even, \(n^3 + 5n\) is even. Test each \(n\): - \(n = 0\): Even - \(n = 1\): Odd - \(n = 2\): Even - \(n = 3\): Odd - \(n = 4\): Even - \(n = 5\): Odd.

Test Divisibility by 3

A number is divisible by 3 if the sum of its digits is divisible by 3. We check \(n^3 + 5n\) modulo 3 for each \(n\): - \(n = 0\): \(0^3 + 5(0) = 0 \equiv 0 \mod 3\) - \(n = 1\): \(1^3 + 5(1) = 6 \equiv 0 \mod 3\) - \(n = 2\): \(2^3 + 5(2) = 18 \equiv 0 \mod 3\) - \(n = 3\): \(3^3 + 5(3) = 54 \equiv 0 \mod 3\) - \(n = 4\): \(4^3 + 5(4) = 84 \equiv 0 \mod 3\) - \(n = 5\): \(5^3 + 5(5) = 150 \equiv 0 \mod 3\).

Combine Conditions for n to be divisible by 6

We combine the results of divisibility by 2 and by 3. Those \(n\) which satisfy both conditions are: - \(n = 0\): Divisible by both 2 and 3 - \(n = 2\): Divisible by both 2 and 3 - \(n = 4\): Divisible by both 2 and 3.

Key concepts

Divisibility Rules

Understanding divisibility rules makes checking whether one number divides another much simpler. These rules are like shortcuts that save you time.

• Divisibility by 2: A number is divisible by 2 if it is even. This means the last digit of the number is 0, 2, 4, 6, or 8.
• Divisibility by 3: A number is divisible by 3 if the sum of all its digits is divisible by 3.
• Divisibility by 6: A number is divisible by 6 only if it is divisible by both 2 and 3. So, a number must be even and the sum of its digits should be divisible by 3 at the same time.

In our problem, we need to check both conditions for divisibility by 6. Once you practice these rules with a few examples, you'll be able to identify divisible numbers more quickly. They can be handy in solving many mathematical problems. Simple arithmetic with these rules can also give insights before using more involved methods.

Modulo Operation

The modulo operation helps determine the remainder after division. It's a handy tool for checking divisibility and simplifies complex calculations greatly. When we say "a modulo b" (written as \(a \equiv c \pmod{b}\)), we are interested in the remainder when dividing a by b.
For example, if you calculate \(18 \div 3\), the remainder is 0 because 18 divides by 3 evenly. Therefore, \(18 \equiv 0 \pmod{3}\). If you instead take \(17 \div 3\), the remainder is 2 thus, \(17 \equiv 2 \pmod{3}\).
In our exercise, we used modulo 2 and modulo 3 to examine if the polynomial \(n^3 + 5n\) is divisible by these numbers:

• If \(n^3 + 5n \equiv 0 \pmod{2}\), it means the result is even, satisfying one of the divisibility conditions for 6.
• Likewise, if \(n^3 + 5n \equiv 0 \pmod{3}\), the result is divisible by 3.

By combining conditions with modulo operation, it allowed us to test each candidate \(n\) straightforwardly.

Problem Solving Steps

Solving mathematical problems systematically can help break down what seems complex into more straightforward tasks. Let's take an example from our exercise to see how structuring your approach can lead to solutions:

• Understand the Problem: Begin by clearly understanding what you're supposed to find. In our case, it was elements \(n\) such that \(n^3 + 5n\) is divisible by 6.
• Break Down the Requirements: Recognize divisibility by 6 requires divisibility by both 2 and 3, prompting us to check these separately.
• Test Each Condition: Methodically test all possible values in the given set for these conditions. For example, see which values of \(n\) make \(n^3 + 5n\) even, and which make it divisible by 3.
• Combine Findings: Find those values satisfying all criteria. Here, only \(n = 0, 2,\) and \(4\) met both divisibility conditions.

Following these steps doesn't just answer the exercise; it builds confidence and understanding in approaching any similar problem. Each step serves as leverage to reach the next, and before you know it, a seemingly complicated problem has been tackled. \( \)