Problem 47 - Algebra 2

Question

Find a counterexample for each statement. \(5^{n}+1\) is divisible by 6

Step-by-Step Solution

Answer

A counterexample is \( n = 2 \).

Step 1: Understand the Statement

The statement claims that the expression \( 5^n + 1 \) is divisible by 6 for all integer values of \( n \). To find a counterexample, we need to find an integer \( n \) for which \( 5^n + 1 \) is not divisible by 6.

Step 2: Check Simplest Cases

Let's start checking simple cases, beginning with small integer values for \( n \). This will help us identify a case where the statement might fail.

Step 3: Evaluate Different Values of n

- For \( n = 1 \), we have \( 5^1 + 1 = 6 \). 6 is divisible by 6.- For \( n = 2 \), we have \( 5^2 + 1 = 25 + 1 = 26 \). 26 divided by 6 gives a remainder of 2, not divisible by 6.Thus, \( n = 2 \) is a counterexample.

Step 4: Validate the Counterexample

We have found that \( 5^2 + 1 = 26 \), and when dividing 26 by 6, the remainder is 2. This means that 26 is not divisible by 6, confirming that \( n = 2 \) is indeed a counterexample.

Key Concepts

Integer Exponentiation

Integer exponentiation refers to raising an integer to the power of another integer. When you see a number like \(5^n\), it represents multiplying the base, which is 5, by itself \(n\) times. This mathematical operation is fundamental and widely used:

• For \(n = 1\), \(5^1 = 5\).
• For \(n = 2\), \(5^2 = 25\).
• For \(n = 3\), \(5^3 = 125\).

As the exponent increases, the result grows quickly. In the provided exercise, determining the divisibility of expressions involving exponentiation like \(5^n + 1\) requires careful evaluation of how output expressions relate to division rules. Understanding integer exponentiation is crucial to explore further into algebraic expressions.

Divisibility

Divisibility is a key concept in arithmetic where one integer can be divided by another without leaving a remainder. For example, 12 can be divided by 3, as \(12 \div 3 = 4\), and the remainder is 0. Here, we explore whether \(5^n + 1\) is divisible by 6.
The statement becomes invalid if you find a value that results in a remainder when divided by 6. A counterexample like \(5^2 + 1 = 26\) is not divisible by 6 because dividing 26 by 6 leaves a remainder of 2. Recognizing numbers that do not fit the divisibility criteria helps in understanding how exceptions occur and why certain numbers are not divisible by others.

Algebraic Expressions

An algebraic expression involves numbers, variables, and arithmetic operations like addition, subtraction, multiplication, and exponentiation. Expressions like \(5^n + 1\) illustrate this by combining variables and operations:

• \(5^n\) involves exponentiation of a variable \(n\).
• The expression \(5^n + 1\) adds 1 to the result from the exponentiation.

When analyzing such expressions, you determine how changes in variables affect the overall expression. For example, changing \(n\) alters the result of the expression, which matters when you check for divisibility. Understanding algebraic expressions allows for a better grasp of the mathematical structure and behavior of various mathematical statements.

Number Theory

Number theory is a branch of pure mathematics focused on the properties and relationships of numbers, particularly integers. It involves concepts like divisibility, prime numbers, and integer solutions to equations. Here, number theory helps in examining divisibility within expressions like \(5^n + 1\).
Considering statements about divisibility helps identify patterns and the behavior of numbers in equations. As illustrated through finding counterexamples such as \(n = 2\), number theory helps confirm or refute general claims about numbers. This branch of mathematics is fundamental for understanding why some mathematical statements hold true across all integers while others do not.