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Beginning Activity Beginning Activity 2: Prime Factorizations
A4US

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. 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.)

1.

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=22⋅3⋅5, we say that 22⋅3⋅5 is a prime factorization of 60.

2.

Write the number 40 as a product of prime numbers by first writing 40=2⋅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⋅8 and then factoring 8 into a product of primes.

3.

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.

4.

Repeat Exercise 2 and Exercise 3 with 150. First, start with 150=3⋅50, and then start with 150=5⋅30.