Problem 55 - Algebra 1: Concepts and Skills

Question

Find the number with the given prime factorization. (Skills Review p. 761) $$ 2 \cdot 3 \cdot 11 $$

Step-by-Step Solution

Answer

The number with the prime factorization of 2, 3, 11 is 66.

Step 1: Identify the Prime Factors

The given prime factors are 2, 3, and 11.

Step 2: Multiply the Prime Factors

The next step is to multiply these prime factors together. Therefore, compute \(2 \cdot 3 \cdot 11\).

Step 3: Compute the Result

The result of multiplying 2, 3, and 11 is 66.

Key Concepts

Multiplying Prime Factors

Multiplying prime factors is like putting together building blocks to form a complete structure. When given a set of prime numbers, we can find their product to gain the original number they were factored from. It's particularly straightforward because all we have to do is multiply the numbers together,
as shown in our exercise with the factors 2, 3, and 11. This is how it unfolds:

• Start with the first two prime numbers in the sequence: 2 and 3.
• Multiply these together to get 6.
• Next, take the result of 6 and multiply it by the next prime number, which is 11.
• The result is 66, which is the original number.

By multiplying the prime factors, we reverse the process of breaking down a number into its smallest part, reassembling it back into the integer it was before. This skill is foundational in mathematics, aiding in understanding larger concepts like greatest common divisors and least common multiples.

Prime Numbers

Prime numbers are the building blocks of all numbers. They are unique in that they have exactly two distinct positive divisors: 1 and themselves. This means they cannot be divided evenly by any other number. For example, the prime numbers in our exercise—2, 3, and 11—are such that they each only have divisors of 1 and themselves. Here's a quick look at these prime properties:

• 2 is the smallest and the only even prime number.
• 3 follows as the second smallest prime, forming a basic prime duo with 2.
• 11 is further down the list of primes and demonstrates larger prime characteristics.

The significance of these numbers in factorization is profound. Since every integer greater than 1 can be expressed as a product of primes, they form the core of integer decomposition. Understanding primes is essential to mastering factorization and many other arithmetic operations.

Integer Factorization

Integer factorization involves breaking down a number into its prime components—its most basic building blocks. This process helps us understand the fundamental structure of numbers. For the number 66, as seen in the exercise, integer factorization is what gave us the prime factors of 2, 3, and 11.

• Begin the process by starting with the smallest prime number, and check if it divides the given number evenly.
• If it does, divide and continue testing with the resulting quotient, using each next potential prime.
• Repeat these steps until the quotient is a prime number itself, ensuring no more division is possible.

Factoring integers is a key aspect of number theory, with applications in cryptography, coding theory, and beyond. Through factorization, otherwise complex numbers are broken into simple, digestible parts, highlighting the relevance of the prime 'building blocks' in mathematics.