Beginning Activity Beginning Activity 2: Prime Factorizations
Recall that a natural number \(p\) is a prime number provided that it is greater than 1 and the only natural numbers that divide \(p\) are 1 and \(p\text{.}\) A natural number other than 1 that is not a prime number is a composite number. The number 1 is neither prime nor composite. (See Activity 8 from Section 2.4.)
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Give examples of four natural numbers that are prime and four natural numbers that are composite.
Theorem 4.11 in Section 4.2 states that every natural number greater than 1 is either a prime number or a product of prime numbers.
When a composite number is written as a product of prime numbers, we say that we have obtained a prime factorization of that composite number. For example, since \(60 = 2^2 \cdot 3 \cdot 5\text{,}\) we say that \(2^2 \cdot 3 \cdot 5\) is a prime factorization of 60. Write the number 40 as a product of prime numbers by first writing \(40 = 2 \cdot 20\) and then factoring 20 into a product of primes. Next, write the number 40 as a product of prime numbers by first writing \(40 = 5 \mspace{1mu}\cdot\mspace{1mu} 8\) and then factoring 8 into a product of primes. In Exercise 2, we used two different methods to obtain a prime factorization of 40. Did these methods produce the same prime factorization or different prime factorizations? Explain. Repeat Exercise 2 and Exercise 3 with 150. First, start with \(150 = 3 \cdot 50\text{,}\) and then start with \(150 = 5 \cdot 30\text{.}\)2.
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