# MAT 311 Number Theory

### Homework:

Problems marked with an asterisk (*) are for extra credit.
• HW 1 (due 02/05 in class) [solutions]
• Chapter 1: Ex 1.1, 1.3, 1.4
• Chapter 2: Ex 2.1, 2.3 a) and b), 2.4
• HW 2 (due 02/12 in class) [solutions]
• Chapter 3: Ex 3.1, 3.2, 3.3
• Chapter 4: Ex 4.2 a), b), d) only
• Chapter 5: Ex 5.1, 5.3, 5.5 a) and b) only
• HW 3 (due 02/19 in class) [solutions]
• Chapter 6: Ex 6.1, 6.2, 6.3, 6.4 a) only
• Chapter 7: Ex 7.2, 7.3, 7.4
• HW 4 (due 02/26 in class) [solutions]
• Find the prime-power factorization of 20!
• How many zeroes are at the end of 50! in decimal notation? Explain!
• Which positive integer numbers have exactly three positive divisors? Which have exactly four positive divisors?
• Show that if a and b are positive numbers such that a3|b2, then a|b.
• Chapter 7: Ex 7.5
• Chapter 8: Ex 8.2, 8.3, 8.4
• HW 5 (due 03/04 in class) [solutions]
• Chapter 9: Ex. 9.1, 9.4
• Chapter 10: Ex 10.1, 10.2, 10.3 (a)
• Chapter 11: Ex 11.1, 11.2
• HW 6 (due 03/11 in class) [solutions]
• Chapter 11: Ex 11.5, 11.8, 11.9, 11.10, 11.11 a)
• What should the check digit be to complete the following ISBN: 0-19-081082 ?
• Determine if the ISBN 1-09-231221-3 is valid?
• HW 7 (due 03/25 in class) [solutions]
• Chapter 19: Ex 19.1, 19.3
• Find the sum of the positive divisors of each of the following integers: 2100, 196, and 20!
• Which positive integers have an odd number of positive divisors?
• Find the smallest positive integer n with τ(n)=3.
• Which positive integers have exactly four positive divisors?
• Show that no two positive integers have the same product of divisors.*
• Find the following values of the Moebius μ function: μ(12), μ(30)
• Show that for any positive integer n we have μ(n) μ(n+1) μ(n+2) μ(n+3)=0.
• Is it possible for the Moebius μ function to vanish for 5 consecutive values of n?
• Use the Moebius inversion formula to express the Euler φ function in terms of the Moebius μ function.
• Chapter 13: Ex 13.3
• HW 8 (due 04/15 in class) [solutions]
• Chapter 17: Ex 17.1, 17.2, 17.4
• Chapter 18: 18.1, 18.2
• Find the primes p and q if pq=4,386,607 and φ(pq)=4,382,136. Explain the method you have used.
• HW 9 (due 04/22 in class) [solutions]
• Chapter 20: Ex 20.1, 20.2, 20.3, 20.4, 20.6, 20.7 a) only, 20.8
• Chapter 21: Ex 21.1, 21.3
• HW 10 (due 04/29 in class)
• Chapter 22: Ex 22.3
• Chapter 23: Ex 23.1, 23.3, 23.5
• Chapter 24: Ex 24.1, 24.2, 24.4, 24.6, 24.7, 24.9
• Chapter 25: Ex 25.2 (bonus problem)
• HW 11 (due 05/6 in class; bonus problems)
• Chapter 25: Ex 25.3, 25.5
• Chapter 26: Ex 26.1, 26.6