MATH 140 InClass Exercise on Primes and Factorization

Names______________________________________________________

1. Find the prime factorizations of each of the following. Write your answers in the following form: .

a) 448 = ____________

b) 2475 = ____________

c) 3757 = ____________

d) 1512 = ____________

2. Recall that d(n) is the number of positive factors of n. For example d(12) = 6, since the positive factors of 12 are {1, 2, 3, 4, 6, 12}. Find d(n) below:

a) d(50) = ____________

b) d(30) = ____________

c) d(144) = ____________

3. Test the following numbers to determine if they are primes. In each case determine the largest prime you have to test with. Answer yes if the number is prime and give a factorization if it is not a prime.

a) 319 _______ largest prime to test___________

b) 321_______ largest prime to test___________

c) 447_______ largest prime to test___________

d) 191 _______ largest prime to test___________

4. Write the numbers from 1 to 20 in base 5. Looking at the list, find a rule for divisibility by 2. That is, find a divisibility test for 2 given that the numbers are expressed in base 5. Note that 6 = 11₅ so the rule from base 10 will not work.

Rule for divisibility by 2:

5. Guess a formula for d(n) when n is a number of the form n = p^aq^b where p and q are primes. To guess the rule compute the following examples and look for a pattern.

n	2^13^1	2^23^1	2^23^2	2^33^2	2^33^3	3^15^1	2^63^1	5^17^1
d(n)