Problem 89 - Fundamentals of Mathematics

Question

For the following problems, reduce, if possible, each of the fractions to lowest terms. $$\frac{39}{13}$$

Step-by-Step Solution

Answer

The fraction \( \frac{39}{13} \) reduces to 3.

Step 1: Identify the Numerator and Denominator

The given fraction is \( \frac{39}{13} \). Here, the numerator is 39, and the denominator is 13.

Step 2: Find the Greatest Common Divisor (GCD)

To reduce a fraction to its lowest terms, we need to divide both the numerator and the denominator by their greatest common divisor (GCD). The GCD of 39 and 13 can be found by determining the largest number that divides both 39 and 13 without leaving a remainder. Since 13 is a prime number and it divides 39 evenly, the GCD is 13.

Step 3: Divide the Numerator and Denominator by the GCD

Divide both the numerator and the denominator by the GCD found in the previous step. This gives us: \[ \frac{39}{13} = \frac{39 \div 13}{13 \div 13} = \frac{3}{1} \].

Step 4: Interpret the Result

After dividing both terms by the GCD, the fraction becomes \( \frac{3}{1} \), which further simplifies to 3. Therefore, the fraction \( \frac{39}{13} \) reduces to the whole number 3.

Key Concepts

Lowest Terms

Reducing a fraction to its lowest terms means expressing it in the simplest form possible. In other words, the numerator and denominator should have no common divisors other than 1. This process helps in making fractions easier to work with. To reduce a fraction:

• Identify the numerator and the denominator.
• Find their greatest common divisor (GCD).
• Divide both the numerator and the denominator by the GCD.

For example, the fraction \( \frac{39}{13} \) can be reduced to its lowest terms by dividing both the numerator and denominator by 13, resulting in \( \frac{3}{1} \), which simplifies to 3.

Greatest Common Divisor

The greatest common divisor (GCD) is the largest number that divides two or more numbers without leaving a remainder. Finding the GCD is crucial for reducing fractions to their lowest terms. Different methods can be used to find the GCD:

• Prime Factorization: Break down the numbers into their prime factors and multiply the common factors.
• Division Method: Keep dividing the larger number by the smaller and use the remainder to repeat the process until zero is left; the last non-zero remainder is the GCD.
• Euclidean Algorithm: An efficient way using division and remainders to find the GCD.

In the example of \( \frac{39}{13} \), since 13 divides 39 exactly, the GCD is 13. After finding the GCD, you divide both parts of the fraction by this number to simplify it.

Prime Numbers

Prime numbers are numbers greater than 1 that have no divisors other than 1 and themselves. They are fundamental in the study of numbers because they are the building blocks of whole numbers. Understanding prime numbers helps significantly in finding the GCD of two numbers, as well as in simplifying fractions.

• Examples of prime numbers include 2, 3, 5, 7, 11, 13, etc.
• Every whole number greater than 1 is either a prime number or a product of prime numbers.

In the fraction \( \frac{39}{13} \), 13 is a prime number, making it straightforward to determine that it evenly divides 39, assisting in quickly finding the GCD. Recognizing these characteristics can speed up the process of reducing fractions.