Problem 18 Find a counterexample for each s... [FREE SOLUTION]

Chapter 11: Problem 18

Find a counterexample for each statement. $$ 3^{n}+1 \text { is divisible by } 4 $$

Short Answer

Expert verified

A counterexample is $n = 2$ since $3^2 + 1 = 10$ is not divisible by 4.

Step by step solution

Understand the Statement

The statement claims that the expression $3^n + 1$ is divisible by 4 for any integer $n$. To find a counterexample, we need to find an $n$ for which this expression is NOT divisible by 4.

Choose a Value for n

Let's test $n = 2$. This is a small and simple number to calculate and verify.

Calculate 3^n + 1 for n = 2

Substitute $n = 2$ into the expression: $3^2 + 1 = 9 + 1 = 10$.

Check Divisibility by 4

Now check if $10$ is divisible by 4. Divide $10$ by 4. The quotient is $2$ with a remainder of $2$, meaning $10$ is not divisible by 4.

Conclude with the Counterexample

Since $10$ is not divisible by 4, $n = 2$ serves as a counterexample, disproving the original statement.

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Key Concepts

These are the key concepts you need to understand to accurately answer the question.

Divisibility

Divisibility is a fundamental concept in mathematics that involves determining whether one integer can be evenly divided by another. To say that a number $ a $ is divisible by another number $ b $ means that dividing $ a $ by $ b $ leaves no remainder.
For example, $ 12 $ is divisible by $ 4 $ because $ 12 \div 4 = 3 $ with no leftover. It鈥檚 like sharing items equally among friends; if you can do so without any leftover items, you have divisibility.
To check if a number is divisible by another:

Divide the numbers.
Check the remainder.
If there's no remainder, it's divisible.

This concept is often used to simplify problems, find patterns, or make calculations easier in more complex mathematics.

Integer Exponentiation

Integer exponentiation refers to repeatedly multiplying a base number by itself a certain number of times, denoted by its exponent. If a number $ a $ is raised to the power of $ n $, it can be written as $ a^n $.
To calculate it, you multiply $ a $ by itself $ n-1 $ more times.
For instance, if $ a = 3 $ and $ n = 2 $, then $ 3^2 = 3 \times 3 = 9 $.
It鈥檚 helpful to remember that:\

Any number raised to the power of $ 0 $ is $ 1 $.
Any number raised to the power of $ 1 $ is the number itself.

Exponentiation is a powerful tool in algebra and various fields, helping in equations, growth models, and many aspects where repeated multiplication is involved.

Modulus Operations

The modulus operation, commonly known as modulo, finds the remainder of a division of two numbers. Symbolically, it is expressed as $ a \mod b $ which equals the remainder when $ a $ is divided by $ b $.
For example, $ 10 \mod 4 $ equals $ 2 $, because when you divide $ 10 $ by $ 4 $, the remainder is $ 2 $.
Key points about modulus:

It helps determine divisibility.
Useful in programming, cryptography, and creating loops that repeat every few iterations.
Used to check if a number is even or odd, since any even number $ n $ gives $ n \mod 2 = 0 $.

Understanding module operations aids in solving puzzles, optimizing processes, and breaking down complex mathematical problems.

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91影视

Find a counterexample for each statement. $$ 3^{n}+1 \text { is divisible by } 4 $$

Short Answer

Step by step solution

Understand the Statement

Choose a Value for n

Calculate 3^n + 1 for n = 2

Check Divisibility by 4

Conclude with the Counterexample

Key Concepts

Divisibility

Integer Exponentiation

Modulus Operations

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