Problem 18 - Precalculus Mathematics for Calculus

Question

Show that \(n^{3}-n+3\) is divisible by 3 for all natural numbers \(n\)

Step-by-Step Solution

Answer

The expression \(n^3 - n + 3\) is divisible by 3 for all natural numbers \(n\).

Step 1: Analyze divisibility by 3

To show that \(n^3 - n + 3\) is divisible by 3, we need to prove that it results in a multiple of 3 for every natural number \(n\).

Step 2: Use modular arithmetic

Using modular arithmetic, we check \(n^3 - n + 3\) modulo 3 for \(n \equiv 0, 1, 2 \pmod{3}\). This will show if the expression is congruent to 0 for each case.

Step 3: Check case \(n \equiv 0 \pmod{3}\)

For \(n \equiv 0 \pmod{3}\), substitute in the expression: \(n^3 - n + 3 \equiv 0^3 - 0 + 3 \equiv 3 \equiv 0 \pmod{3}\). The expression is 0 modulo 3.

Step 4: Check case \(n \equiv 1 \pmod{3}\)

For \(n \equiv 1 \pmod{3}\), substitute in the expression: \(n^3 - n + 3 \equiv 1^3 - 1 + 3 \equiv 1 - 1 + 3 \equiv 3 \equiv 0 \pmod{3}\). The expression is 0 modulo 3.

Step 5: Check case \(n \equiv 2 \pmod{3}\)

For \(n \equiv 2 \pmod{3}\), substitute in the expression: \(n^3 - n + 3 \equiv 2^3 - 2 + 3 \equiv 8 - 2 + 3 \equiv 9 \equiv 0 \pmod{3}\). The expression is 0 modulo 3.

Step 6: Conclusion

Since \(n^3 - n + 3\) results in 0 modulo 3 for \(n \equiv 0, 1, 2 \pmod{3}\), the expression is divisible by 3 for all natural numbers \(n\).

Key Concepts

Divisibility Rules

Divisibility rules are a set of simple guidelines that help us determine whether a number can be evenly divided by another number without using long division. These rules can greatly simplify arithmetic calculations. One commonly used rule is the divisibility rule for 3. To check if a number is divisible by 3, we can use modular arithmetic. For any integer, if the sum of its digits is a multiple of 3, then the number itself is divisible by 3.

In modular arithmetic, when we say a number is congruent to another number modulo 3, it means the difference between them is divisible by 3. This assists in confirming divisibility for expressions like polynomial functions.

• For instance, the rule can quickly tell us that 123 (since 1 + 2 + 3 = 6, a multiple of 3) is divisible by 3.
• Using this understanding can help verify expressions without performing extensive computation.

Natural Numbers

Natural numbers are the basic building blocks of mathematics. These numbers start from 1 and increase infinitely, including every whole number that follows it, like 2, 3, 4, and so on. They do not include zero or any negative numbers.

Natural numbers are often used when counting objects, as they perfectly handle quantities of items. They also play a significant role in numerous mathematical functions, equations, and problems, including polynomial expressions.

• In the context of the given problem, we deal with natural numbers as the value of the variable \(n\).
• Because natural numbers are infinite, proving divisibility for all natural numbers involves proving the pattern holds true irrespective of the particular value of \(n\).

Polynomial Functions

Polynomial functions are expressions that involve a sum of powers of variables multiplied by coefficients, such as \(a_nx^n + a_{n-1}x^{n-1} + \ldots + a_1x + a_0\). These functions encompass a wide range of forms, including linear, quadratic, cubic, and higher degrees.

In the specific problem investigated here, we focus on the polynomial function \(n^3 - n + 3\). A key point of interest in dealing with polynomials like this is their behavior under various mathematical operations, including divisibility and modular arithmetic.

• Cubic functions like \(n^3\) can have complex behavior, but examining them modulo operations can simplify analysis and proofs of divisibility.
• Using patterns seen in lower degrees like squares or lines, we analyze cubic polynomials to establish consistency across all natural numbers.