Question

Find the least common denominator. $$\frac{1}{16}, \frac{9}{20}$$

Short answer

The Least Common Denominator is 80.

Step-by-step solution

Prime Factorization

The first step is to perform prime factorization on each of the denominators. For 16, the prime factors are \(2 \times 2 \times 2 \times 2 \) and for 20, the prime factors are \(2 \times 2 \times 5\).

Identifying the Common and Additional Factors

The common prime factors between 16 and 20 are two 2s. However, there are additional two 2s in the prime factorization of 16, and one 5 in the prime factorization of 20.

Determining the LCD

The LCD is obtained by multiplying the common prime factor(s) and the extra prime factor(s) from each denominator. Thus, the LCD = \(2 \times 2 \times 2 \times 2 \times 5 = 80\).

Key concepts

Prime Factorization

Prime factorization is the process of breaking down a number into its basic building blocks, which are called prime numbers. Prime numbers are those that can only be divided by 1 and themselves. They are the 'atoms' of the mathematical world, as they cannot be divided into smaller number factors.

For students grappling with this concept, imagine you are dismantling a complex toy into its smallest indestructible pieces. Those pieces (prime numbers) can be combined in various ways to rebuild any number. For example, to find the prime factors of 16, you would divide it by the smallest prime number, which is 2, and continue dividing until you can't divide any further. The end result for 16 is \(2 \times 2 \times 2 \times 2\), as shown in the textbook solution.

Common Factors

When we talk about common factors, we are referring to those prime factors that different numbers share. It's like finding common traits among a group of friends. In math, we list out the prime factors of each number and then spot the ones that appear in all lists. These are the factors they have in common.

To illustrate, consider our numbers 16 and 20 from the exercise. As explained, 16 is made of four 2s while 20 is made of two 2s and a 5. They share two 2s — these are their common factors. Understanding the common factors is vital when looking to add, subtract, or compare fractions, as well as finding the least common denominator (LCD).

LCD Determination

The Least Common Denominator (LCD) is simply the smallest multiple that two or more fractions share as a denominator. Think of it as scheduling a meeting when everyone is available. To find it, you take the prime factors you've identified, both common and unique, and multiply them together.

In our example, we take the common factors of 16 and 20, which are two 2s, and then multiply them by the additional two 2s from 16 and the one 5 from 20 to find the least common multiple (LCM), which doubles as our LCD: \(2 \times 2 \times 2 \times 2 \times 5 = 80\). Thus, 80 is the number that can be used as a common denominator for fractions with denominators of 16 and 20, ensuring that no information is lost and the fractions can be compared or combined accurately.