Problem 91 - Algebra 1: Concepts and Skills

Question

Write the prime factorization of the number if it is not a prime number. If a number is prime, write prime. $$ 20 $$

Step-by-Step Solution

Answer

The prime factorization of 20 is \(2^2 * 5\).

Step 1: Verification of Prime Number

Verify if the given number 20 is a prime number. A prime number has only two distinct divisors: 1 and itself. In case of 20, it has more than two divisors (1, 2, 4, 5, 10, 20), so it is not a prime number.

Step 2: Prime Factorization

Divide the number by the smallest prime number, i.e., 2. The quotient is 10. Continue this process with the quotient until the quotient is a prime number. For 20, it goes like this: 20/2 = 10, 10/2 = 5. Now, 5 is a prime number. Hence, the process is stopped.

Step 3: Writing the Result

Write down the prime factorization of the number using the prime numbers obtained in the above steps. For 20, the prime factors are 2, 2, and 5. So, the prime factorization can be written as: \(20 = 2^2 * 5\).

Key Concepts

Prime Numbers

Prime numbers are unique numbers greater than 1 that have no divisors other than 1 and themselves. In simplest terms, a prime number cannot be divided evenly by any other number except for the number 1 and the number itself.
Here are some important points about prime numbers:

• The first prime numbers are 2, 3, 5, 7, 11, 13, and so on.
• The number 2 is the only even prime number. All other even numbers can be divided evenly by 2, which makes them not prime.
• Prime numbers are the basic building blocks of all numbers. Any whole number greater than 1 is either a prime number or can be formed by multiplying prime numbers together.

Identifying prime numbers is crucial for various mathematical applications, including encryption algorithms, coding theory, and of course, prime factorization.

Divisors

Divisors, in math, are the numbers that divide another number completely without leaving a remainder. Understanding divisors is vital to determine if a number is prime or to perform a prime factorization.
A number like 20, which we are using as an example, has divisors that include 1, 2, 4, 5, 10, and 20 itself. These numbers can divide 20 into whole numbers:

• 1 × 20 = 20
• 2 × 10 = 20
• 4 × 5 = 20

By finding the divisors, we can determine that 20 is not a prime number. Prime numbers only have exactly two divisors. Since 20 has more than two divisors, it gives us insight into how we can further break it down or "factorize" it using prime numbers.

Algebra

Algebra often involves using numbers, letters, and mathematical symbols to represent problems or relationships. It is widely used to make sense of arithmetic operations and extends to complex problems.
Although algebra might seem distant when discussing concepts like prime factorization, it provides a systematic way to express our findings. For instance, let's consider the number 20 again:

• Using algebra, we can express the prime factorization of 20 as an equation: \( 20 = 2^2 \times 5 \).
• This representation helps us understand how 20 is made up of smaller prime numbers multiplied together.

Algebra allows us to manipulate and solve equations by preserving the balance around the equals sign. Thus, it's not just about numbers, but understanding and applying rules to reach solutions efficiently.