Problem 56 - Fundamentals of Mathematics

Question

Write each number as a product of prime factors. 20

Step-by-Step Solution

Answer

20 = 2^2 \times 5

Step 1: Identify the Smallest Prime Number

The number 20 is even, which means it is divisible by the smallest prime number, 2. We start the prime factorization with 2.

Step 2: Divide by 2

Divide 20 by 2: \[ 20 \div 2 = 10 \] This gives us 10.

Step 3: Divide by 2 Again

Continue the process with the result, 10. Since 10 is also even, divide by 2 again: \[ 10 \div 2 = 5 \] This gives us 5.

Step 4: Check for Primality

Now we have 5, which is a prime number. The process stops here as 5 cannot be divided further using prime factors.

Step 5: Write Prime Factorization

Combine all the prime numbers you've used for division to write 20 as a product of its prime factors: \[ 20 = 2 \times 2 \times 5 \] or equivalently \[ 20 = 2^2 \times 5 \]

Key Concepts

Prime Numbers

Prime numbers are the building blocks of all numbers. Think of them as the atoms of our numerical world. These numbers are unique because they have only two divisors: 1 and the number itself. This means you cannot break them down into smaller numbers multiplied together. Some of the most well-known prime numbers are 2, 3, and 5.

• Unique Properties: A prime number has no divisors other than 1 and itself.
• Infinite in Nature: There are infinite prime numbers.
• Basic Examples: The first few prime numbers are 2, 3, 5, 7, 11, and so on.

Prime numbers are especially important in prime factorization. They serve as the fundamental components that when multiplied together, form other numbers. Recognizing prime numbers is key in simplifying and solving many mathematical problems, including the factorization process.

Multiplication

Multiplication is one of the basic arithmetic operations, and it is crucial in the process of prime factorization. Here, multiplication involves combining numbers to get a product. For example, when you multiply 2 by 5, you get 10.
Prime factorization consists of expressing a number as the product of prime numbers. In our exercise, we represented 20 as a product of its prime factors, which is \[ 20 = 2^2 \times 5 \].

• Combination of Primes: Prime factorization involves combining multiple instances of prime numbers through multiplication.
• Exponentiation: If a prime number appears more than once, we can use exponents. For instance, \(2^2\) signifies that 2 appears twice.

Understanding multiplication helps in writing and simplifying the prime factors of a given number. It is a foundational skill that is used in various mathematical applications.

Divisibility

Divisibility refers to the ability of one number to be divided by another without leaving a remainder. In the context of prime factorization, divisibility helps in breaking down a number into its prime components. For example, 20 is divisible by 2 because \[ 20 \div 2 = 10 \].

• Even Numbers: Numbers divisible by 2 are called even. In the exercise, 20 being even indicates 2 as a prime divisor.
• Divisibility Rule for 2: Any even number, or a number ending in 0, 2, 4, 6, or 8, is divisible by 2.
• Checking for Prime Status: Once a number cannot be divided any further except by 1 and itself, it is a prime number.

Using the concept of divisibility, we simplify the process of finding the prime factors. Recognizing which numbers are divisible by smaller primes is an essential skill developed through practicing these kinds of exercises.