Which Expressions Have Exactly Two Factors? Choose ALL That Apply.A. $xy$B. $4 \cdot N \cdot N$C. $a + 7$D. $3(b-1$\]E. $2^5$F. $m^2 + 8$

In mathematics, prime factorization is a process of breaking down a composite number into a product of prime numbers. This concept is crucial in understanding the properties of numbers and their factors. In this article, we will explore which expressions have exactly two factors.

What are Factors?

Factors are numbers that can be multiplied together to get another number. For example, the factors of 12 are 1, 2, 3, 4, 6, and 12. Factors can be prime or composite numbers.

Prime Numbers

Prime numbers are numbers that have exactly two factors: 1 and themselves. For example, the prime number 5 has only two factors: 1 and 5.

Composite Numbers

Composite numbers are numbers that have more than two factors. For example, the number 12 has more than two factors: 1, 2, 3, 4, 6, and 12.

Expressions with Exactly Two Factors

Now, let's analyze the given expressions and determine which ones have exactly two factors.

A. $xy$

The expression $xy$ represents the product of two variables, $x$ and $y$. Since $x$ and $y$ are variables, they can take any value, and the product $xy$ will have more than two factors. Therefore, option A is not correct.

B. $4 \cdot n \cdot n$

The expression $4 \cdot n \cdot n$ can be rewritten as $4n^2$. Since $4$ is a constant and $n$ is a variable, the expression $4n^2$ will have more than two factors. Therefore, option B is not correct.

C. $a + 7$

The expression $a + 7$ represents the sum of a variable $a$ and a constant $7$. Since $a$ is a variable, the expression $a + 7$ will have more than two factors. Therefore, option C is not correct.

D. $3(b-1)$

The expression $3(b-1)$ can be rewritten as $3b - 3$. Since $3$ is a constant and $b$ is a variable, the expression $3b - 3$ will have more than two factors. Therefore, option D is not correct.

E. $2^5$

The expression $2^5$ represents the power of $2$ raised to the exponent $5$. Since $2$ is a prime number and the exponent is a positive integer, the expression $2^5$ will have exactly two factors: $1$ and $2^5$. Therefore, option E is correct.

F. $m^2 + 8$

The expression $m^2 + 8$ represents the sum of a squared variable $m$ and a constant $8$. Since $m$ is a variable, the expression $m^2 + 8$ will have more than two factors. Therefore, option F is not correct.

Conclusion

In conclusion, the expression $2^5$ is the only option that has exactly two factors: $1$ and $2^5$. The other options have more than two factors, making them incorrect.

Key Takeaways

• Prime numbers have exactly two factors: 1 and themselves.
• Composite numbers have more than two factors.
• Expressions with variables will have more than two factors.
• The expression $2^5$ is the only option that has exactly two factors.

Final Answer

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In our previous article, we explored which expressions have exactly two factors. We analyzed five options and determined that only one of them meets the criteria. In this article, we will answer some frequently asked questions related to prime factorization and expressions with exactly two factors.

Q: What is prime factorization?

A: Prime factorization is the process of breaking down a composite number into a product of prime numbers. This concept is crucial in understanding the properties of numbers and their factors.

Q: What are prime numbers?

A: Prime numbers are numbers that have exactly two factors: 1 and themselves. For example, the prime number 5 has only two factors: 1 and 5.

Q: What are composite numbers?

A: Composite numbers are numbers that have more than two factors. For example, the number 12 has more than two factors: 1, 2, 3, 4, 6, and 12.

Q: How do I determine if a number is prime or composite?

A: To determine if a number is prime or composite, you can use the following steps:

1. Check if the number is divisible by any prime numbers less than or equal to its square root.
2. If the number is divisible by any of these prime numbers, it is composite.
3. If the number is not divisible by any of these prime numbers, it is prime.

Q: What are the properties of expressions with exactly two factors?

A: Expressions with exactly two factors have the following properties:

• They are prime numbers or powers of prime numbers.
• They have only two factors: 1 and themselves.
• They cannot be broken down into simpler expressions.

Q: Can you give an example of an expression with exactly two factors?

A: Yes, an example of an expression with exactly two factors is $2^5$. This expression represents the power of 2 raised to the exponent 5 and has exactly two factors: 1 and $2^5$.

Q: What are some common mistakes to avoid when working with prime factorization?

A: Some common mistakes to avoid when working with prime factorization include:

• Assuming that a number is prime without checking its factors.
• Failing to check if a number is divisible by any prime numbers less than or equal to its square root.
• Not considering the properties of expressions with exactly two factors.

Q: How can I apply prime factorization in real-world scenarios?

A: Prime factorization has many real-world applications, including:

• Cryptography: Prime factorization is used to create secure encryption algorithms.
• Coding theory: Prime factorization is used to create error-correcting codes.
• Number theory: Prime factorization is used to study the properties of numbers and their factors.

Conclusion

In conclusion, prime factorization is a fundamental concept in mathematics that has many real-world applications. By understanding the properties of expressions with exactly two factors, you can apply prime factorization in various scenarios. Remember to avoid common mistakes and use the correct techniques to determine if a number is prime or composite.

Key Takeaways

• Prime numbers have exactly two factors: 1 and themselves.
• Composite numbers have more than two factors.
• Expressions with variables will have more than two factors.
• The expression $2^5$ is an example of an expression with exactly two factors.

Final Answer

The final answer is: $\boxed{2^5}$