If an integer n is written as a product of prime numbers, this product (known as its
prime factorization) can be used to determine the number of positive factors of n. For
example, the prime factorization of 28 = 2 × 2 × 7 = 22 × 7
1
. The positive factors of 28
are:
28 = 22 × 7
1
14 = 21 × 7
1
7 = 20 × 7
1
4 = 22 × 7
0
2 = 21 × 7
0
1 = 20 × 7
0
Each positive factor includes 2, 1 or 0 twos, 1 or 0 sevens, and no other prime numbers.
Since there are 3 choices for the number of twos, and 2 choices for the number of sevens,
there are 3 × 2 = 6 positive factors of 28.
(a) How many positive factors does 675 have?
(b) A positive integer n has the positive factors 9, 11, 15, and 25 and exactly fourteen
other positive factors. Determine the value of n.
(c) Determine the number of positive integers less than 500 that have the positive
factors 2 and 9 and exactly ten other positive factors.

a. What are all the factors of 200? b. A prime factor is a factor that is also a prime number what are the prime factors of 200? c. Sometimes it is useful to represent a number as the product of prime factors. How could you write 200 as a product using only prime factors?

Writing a number a as product of only prime numbers is called prime factorization d. Tatiana was writing 200 as a product of prime numbers is called. Prime factorization d. Tatiana was writing 200 as a product of prime numbers. She shared with her team the beginning of her work, which is shown.

What process do you think was going through Tatana's mind when she wrote 200 as a product of prime factors? Do you think it matters what products Tatiana wrote in her second step? What if she wrote 10.20 instead? Finish this prime factorization using Tatiana's process.

7. (a) Find the prime factorization of ged(5!, 2880). [2 marks] (b) How many positive divisors does ged(5!, 2880) have? [2 marks]

Calculate the following limits using the factorization formula , where n is a positive integer and a is a real number. .