Question

According to the Fundamental Theorem of Arithmetic, every natural number greater than 1 can be written as the product of primes in a unique way, except for the order of the factors. For example, \(45=3 \cdot 3 \cdot 5 .\) Write each of the following as a product of primes. (a) 243 (b) 124 (c) 5100

Short answer

(a) 243 = 3^5, (b) 124 = 2^2 × 31, (c) 5100 = 2^2 × 3 × 5^2 × 17.

Step-by-step solution

Prime Factorization of 243

To factorize 243, we start by dividing by the smallest prime number 3: 243 ÷ 3 = 81. Continue dividing by 3: 81 ÷ 3 = 27, 27 ÷ 3 = 9, 9 ÷ 3 = 3, and finally 3 ÷ 3 = 1. Thus, the prime factorization of 243 is: \(243 = 3^5\).

Prime Factorization of 124

Begin by dividing 124 by the smallest prime number 2: 124 ÷ 2 = 62. Continue with 2: 62 ÷ 2 = 31. 31 is a prime number, so it cannot be divided further. Therefore, the prime factorization of 124 is: \(124 = 2^2 \times 31\).

Prime Factorization of 5100

Start by dividing 5100 by the smallest prime number 2: 5100 ÷ 2 = 2550, 2550 ÷ 2 = 1275. Now divide by the next prime number 3: 1275 ÷ 3 = 425. Next, divide by the next prime number 5: 425 ÷ 5 = 85, 85 ÷ 5 = 17. 17 is a prime number. Thus, the prime factorization of 5100 is: \(5100 = 2^2 \times 3 \times 5^2 \times 17\).

Key concepts

Fundamental Theorem of Arithmetic

The Fundamental Theorem of Arithmetic is all about the concept that every natural number greater than 1 is a product of prime numbers. What's interesting though, is once you've found the combination of primes for a number, it will always be the same, regardless of the order of those factors.
For instance, numbers like 45 can be broken down into 3, 3, and 5. Regardless of the order, say 5, 3, 3, it's the same set of prime factors. This idea is crucial for understanding numbers on a deeper level.

Why is this theorem fundamental? It's because it gives us a foundation on which the entire arithmetic system is built. Every time you break down a number into primes, you confirm the unique "building blocks" it is made from. This consistency aids in various mathematical applications, such as simplifying fractions or solving equations.

• Helps simplify math problems.
• Ensures consistent solutions.
• Forms the basis of number theory.

Natural Numbers

Natural numbers are the backbone of basic arithmetic and counting. These are the numbers we use every day, like 1, 2, 3, and so forth. Only positive integers and the number zero fit into this group, and notably, negative numbers and fractions are not called natural numbers.

What's significant about these numbers is their positive and whole characteristics. They form the basis for many numerical concepts. In terms of prime factorization, we only deal with natural numbers greater than 1.

• Includes numbers like 1, 2, 3, 4, etc.
• Used for counting and ordering.
• A foundation for higher-level math concepts.

Natural numbers are important for everyday math, such as buying groceries or counting pennies, and they carry forward into more complex calculations.

Unique Factorization

Unique Factorization ensures that each natural number greater than 1 has one and only one specific set of prime factors. This means that no matter how you rearrange or break down a number, the core set of prime numbers, when multiplied together, will consistently yield the original number's value.

Understanding unique factorization is crucial in algebra and arithmetic because it allows mathematicians to confirm and rely on factor sets for calculations, with no deviation.

• Guarantees one true set of prime factors.
• Helps in simplifying or breaking down numbers.
• Essential for solving equations accurately.

Prime Numbers

Prime numbers are unique because they are greater than 1 and have no divisors other than 1 and themselves. These numbers, like 2, 3, 5, and 7, serve as fundamental units in mathematics because they cannot be formed by multiplying two smaller natural numbers.

Why are prime numbers so vital? They're the building blocks for all other numbers, especially when it comes to generating unique factorization. These numbers help maintain the integrity of numeric expressions and ensure there's only one correct way to express a number as a product of primes.

• Cannot be divided evenly by others except 1 and themselves.
• Used to build and understand other numbers.
• Key in various algorithms and encryption systems.

Understanding prime numbers opens up the door to deeper numerical exploration.