8.1 PROJECT

PRIME FACTORIZATION AND THE NUMBER OF FACTORS

In Section 8.1, you learned about prime and composite numbers. In this project, you will explore a way to determine the number of factors a number has by using its prime factorization.

1. Find the prime factorization of $18$. How many times does each prime factor appear in the prime factorization? Find all possible factors of $18$. How many total factors are there?

2. Find the prime factorization of $100$, and then find all factors of $100$. How many times does each prime factor appear in the prime factorization? How many total factors are there?

It's possible that in parts 1 and 2 you determined the total number of factors by using a brute-force method to list each factor. Let's explore a systematic way to determine the total number of factors.

3. Does any factor in the list of factors of $18$ have prime factors that are not listed in the prime factorization of $18$? Similarly, does any factor in the list of factors of $100$ have prime factors that are not listed in the prime factorization of $100$? Explain why it is or isn't possible for the factors to have a prime factor that is not listed in the prime factorization.

Consider the number $72$, which has a prime factorization of $2\cdot 2\cdot 2\cdot 3\cdot 3$. We can create the factors of $72$ by choosing how many $2$s and $3$s we wish to have in the prime factorization of each individual factor.

4. Find the factors of $72$ that have no $2$s in their prime factorization.

5. Find the factors of $72$ that have exactly one $2$ in their prime factorization.

6. Find the factors of $72$ that have exactly two $2$s in their prime factorization.

7. Find the factors of $72$ that have exactly three $2$s in their prime factorization.

If you combine the factors found in parts 4 through 7 into a list, you will have all of the factors of $72$.

8. How many factors appear in each set of factors in parts 4 through 7? How does this number compare to the number of times $3$ appears in the prime factorization of $72$?

9. How many sets of factors did we find in parts 4 through 7? How does this compare to the number of times $2$ appears in the prime factorization of $72$?

10. Use the answers to parts 8 and 9 to create an algebraic expression to describe how the total number of factors of $72$ compares to the number of $2$s and number of $3$s in the prime factorization of $72$.

11. Can you use the same logic to find an algebraic expression to describe how the total number of factors compares to the prime factorizations of $18$ and $100$? If so, write an expression for each.

12. Find a general rule that gives the number of factors of a given number based on its prime factorization. (Hint: Let $p_1$, $p_2$, and so on represent the prime factors in the prime factorization.)

13. Find the number of factors of $900$ using the prime factorization. Compare using the formula to the brute-force method of listing every factor.