Problem 43 - College Algebra and Calculus: An Applied Approach

Question

Write the prime factorization of the integer.\(2\left(\frac{77}{-11}\right)\)

Step-by-Step Solution

Answer

The prime factorization of \(2\left(\frac{77}{-11}\right)\) is \(-1 \times 2 \times 7\).

Step 1: Simplify the Fraction

First, simplify the fraction \(\frac{77}{-11}\). This simplifies to \(-7\).

Step 2: Multiply by 2

Next, take the result from Step 1, which is \(-7\), and then multiply by 2. This results in \(-14\).

Step 3: Prime Factorization

Finally, take the integer from Step 2, which is \(-14\), and decompose it into prime factors. Because the number is negative, the prime factors are \(-1 \times 2 \times 7\).

Key Concepts

Simplifying Fractions

When faced with a complex fraction, the first step is to simplify it to its lowest terms. Simplifying a fraction involves dividing both the numerator (the top number) and the denominator (the bottom number) by their greatest common factor (GCF). The GCF is the largest number that divides both the numerator and denominator without leaving a remainder.

For example, if we consider the fraction \(\frac{77}{-11}\), we notice that 77 and 11 have a GCF of 11. Dividing both by 11, we get \(\frac{77 \div 11}{-11 \div 11}\) which simplifies to \(\frac{7}{-1}\). Since dividing a number by 1 yields the number itself, and a negative sign in the denominator can be moved to the numerator, the simplified form of this fraction is -7.

Integers

Integers are the set of whole numbers that include all the positive natural numbers (1, 2, 3, ...), zero (0), and the negatives of the natural numbers (-1, -2, -3, ...). In other words, they represent a range that stretches infinitely in both the positive and negative directions, but do not include fractions or decimals.

When working with integers, it is important to remember that multiplication or division by a positive number preserves the sign of the initial integer, while doing so by a negative number flips the sign. In the exercise \(2\left(\frac{77}{-11}\right)\), after simplifying the fraction, we are left with an integer multiplied by 2. Since -7 is a negative integer and 2 is a positive integer, their product -14 is also an integer, specifically a negative one.

Prime Numbers

Prime numbers are the building blocks of whole numbers. A prime number is defined as a natural number greater than 1 that has no divisors other than 1 and itself. This means they cannot be formed by multiplying two smaller natural numbers. The smallest prime numbers are 2, 3, 5, 7, 11, and so on.

When breaking down an integer into its prime factors, like in the prime factorization of -14 from our exercise, we find the prime numbers that, when multiplied together, give us the original integer. For -14, we factor out the negative sign as -1 (considered a unit, not a prime number), and then continue with the prime factors, finding that -14 can be expressed as -1 times 2 times 7, with 2 and 7 being prime numbers. This process is essential for tasks such as simplifying fractions further, finding the least common denominators, or solving problems in different areas of mathematics and number theory.