MATH 140 Class Exercise on Factorization Name__________________________

The goal of this exercise is to find a formula for the function d(n), which is defined by: d(n) equals the number of positive factors of the integer n. For example d(6) = 4 since {1,2,3,6} is the set of all positive factors of 6.

1. Fill in the following chart: for each number n find the number of positive factors of n.

n	1	2	3	4	5	6	7	8	9
d(n)
n	10	11	12	13	14	15	16	17	18
d(n)
n	19	20	21	22	23	24	25	26	27
d(n)

a) What do you notice about d(n) if n is a prime? ____________________

b) When is d(n)an odd number? __________________________

2. Find d(n) for the following set of numbers given by prime factorization:

n	2^0	2^1	2^2	2^3	2^4	2^5
d(n)

n	3^0	3^1	3^2	3^3	3^4	3^5
d(n)

a) Suppose that n = 2^k, what is d(n) = d(2^k) = _______________?

b) Does your answer in a) change if the 2 is replaced by 3? __________

c) Find a number n such that d(n) = 10. ____________

3. Find d(n) for the following set of numbers given by prime factorization.

n	2^13^1	2^15^1	2^17^1	3^15^1	3^17^1	5^17^1	2^23^1
d(n)
n	2^25^1	2^13^2	3^25^1	2^23^2	2^25^2	2^33^1	2^43^1
d(n)

a) Does d(n) depend on the primes or the powers in the factorization? _______________

b) Based on your results what should d(2^a3^b) be if a and b are integers? ___________

c) Does the same rule seem to hold for primes other than 2 and 3? ________

4. Again, find d(n).

n	2^13^15^1	2^13^17^1	2^23^15^1	2^13^15^17^1
d(n)

Suppose that is the prime factorization of n where the p_i are distinct primes. What is d(n)? _______________