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Chapter 5: Problem 100

Factor the integer 35 as the product of the negatives of two prime numbers.

Short Answer

Expert verified
-5 and -7

Step by step solution

01

Understand the Problem

The task is to express the integer 35 as the product of two negative prime numbers.
02

Identify the Prime Factors of 35

The prime factors of 35 are 5 and 7. Thus, 35 can be written as the product of 5 and 7: \[ 35 = 5 \times 7 \].
03

Convert Prime Factors to Negative

Convert both prime factors to their negative counterparts: \[ 35 = (-5) \times (-7) \]
04

Verify the Product

Verify that the product of -5 and -7 is indeed 35: \[ (-5) \times (-7) = 35 \]

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Key Concepts

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

negative prime numbers
Prime numbers are integers greater than 1 that have no divisors other than 1 and themselves. Examples include 2, 3, 5, and 7. In this exercise, we deal with negative prime numbers. Negative prime numbers are simply the negative counterparts of original prime numbers. For example, -2, -3, -5, and -7 are all negative primes. It鈥檚 important to understand that the fundamental properties of these numbers do not change鈥攖hey still have only two distinct divisors: 1 and the number itself, albeit in their negative form. When factoring integers such as 35, we use negative primes to arrive at the same product as we would with positive primes.
integer factorization
Integer factorization is the process of breaking down a composite number into a product of smaller integers, specifically prime factors. For example, 35 is a composite number since it can be written as a product of smaller integers. To factorize 35, we look for prime numbers that, when multiplied together, give 35. In positive terms, these primes are 5 and 7 because \( 5 \times 7 = 35 \). This is the basis of integer factorization. When tasked with expressing 35 as a product of negative primes, we convert 5 and 7 to -5 and -7, thus \( (-5) \times (-7) = 35 \). This conversion to negative primes is simply a step to meet the specific requirements of the problem.
verification of products
Verification of products ensures that the factored form of a number is accurate. After factorizing 35 as \( (-5) \times (-7) \), we need to verify the result. Verification involves re-multiplying the factors to check if they give the original number. In this case, \( (-5) \times (-7) = 35 \). Since the product is the same as the original number, our factorization is correct. Verification serves as a critical step in confirming the accuracy of factorization and helps prevent errors. Thus, in this exercise, we see that converting positive primes to their negative counterparts and then verifying the result is a valid and accurate procedure for integer factorization.

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Most popular questions from this chapter

Factor each polynomial. (Hint: As the first step, factor out the greatest common factor.) \(4 t^{2}(k+9)^{7}+20 t s(k+9)^{7}+25 s^{2}(k+9)^{7}\)

Factor each trinomial. \(-2 a^{2}-5 a b-2 b^{2}\)

Which of the following are differences of cubes? A. \(9 x^{3}-125\) B. \(x^{3}-16\) C. \(x^{3}-1\) D. \(8 x^{3}-27 y^{3}\)

Identify each monomial as a perfect square, a perfect cube, both of these, or neither of these. (a) \(4 x^{3}\) (b) \(8 y^{6}\) (c) \(49 x^{12}\) (d) \(81 r^{10}\) (e) \(64 x^{6} y^{12}\) (f) \(125 t^{6}\)

Factor each binomial completely. \(27 t^{3}-64 s^{6}\)
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