Homework for Math 471 Spring 2026

SHOW ALL YOUR WORK!!!

• Assignment 1   (Due Thursday, February 12)
  • Section 1.3 page 26:   4, 10, 13, 17.
  • Section 1.5 page 37: 5, 10.
  • Section 2.1 page 49: 1, 4, 11.
  • Section 2.2 page 63: 7, 9.
  • Scanned copy of Sec. 1.3
  • Scanned copy of Sec. 1.5
  • Scanned copy of Sec. 2.1
  • Scanned copy of Sec. 2.2
• Assignment 2   (Due Tuesday, February 17)
  • Section 3.1 page 96:   12, 13, 15, 26 (by induction), 27(d).
  • Section 3.2 page 101:   15, 17, 21
  • Section 3.3 page 115:   11 (for c, use p_{pi(n)}<= n < p_{pi(n+1)}), 13.
  • Section 3.4 page 121:   5, 12, 23, 24.
  • Section 3.5 page 133:   15, 16, 28
  • Section 3.6 page 143:   3.
  • Section 4.1 page 153:   1, 2.
  • Section 5.1 page 189:   4, 5, 9, 11.
  • Section 5.2 page 199:   5 (use the extended Euclidean Algorithm), 9, 13.
  • Scanned copy of some of the homework problems (by now I expect you to have the text and will stop posting scans)
  • Solution to some problems and to 3.1.13.
• Assignment 3   (Due Thursday, February 26)
  • Section 5.3 page 203:   1a,b, 11 (Use the extended Euclidean Algorithm), 16, 17, 18, 23.
  • Section 5.4 page 210:   10, 14, 15, 18, 28, 29, 31.
  • Hint for Section 5.4 problem 28:   Show first that there exists a (unique) solution (x,y), with 0< x <= b.
  • Solution to 5.1.11, 5.2.13, 5.3.17. Solution to 5.4.15
  • Section 6.1 page 223:   2(b), 4, 6, 11, 20.
  • Section 6.2 page 234:   1(h), 3, 11, 20, 27.
  • Hint for 27: Proof by contradiction. Assume that x is rational but not an integer, write x=a/b with gcd(a,b)=1, then clear the denominator in the equation and consider a prime factor p of b. You will need to use Problem 6.1.11.
• Assignment 4   (Due Thursday, March 5)
  • Section 7.1 page 250:   3(c), 8(a,b), 10 (prove your answer), 19 (b, d, f, i, l), 20 (b, d, f, i, l), 22.
  • Solution to 5.4.28, 6.1.11, 6.2.11, 6.2.27, 7.1.10.
  • Section 7.2 page 262:   3(a,e), 4(a,e), 5(a,e), 12, 18, 29.
  • Section 7.4 page 283:   7, 10(b), 12, 13,15,21, 25, 29, 31, 36, 37.
  • Extra problem for Section 7.4:   Use the Chinese Remainder Theorem to solve Problem 33 page 266 (in Section 7.2). Hint: Note that 2^{14} is congruent to 1 modulo 43.
  • Section 8.1 page 309:   4, 6a, 9(e,f).
  • Solution to 7.2.5a, 7.2.29, 7.2.33 (using the C.R.T), and 7.4.12.
• Assignment 5   (Due Thursday, March 12)
  • Section 8.2 page 320:   7, 8c, 14, 16, 17.
  • Section 8.3 page 336:   1a, 2a, 3a, 7(a,d),9, 10, 11, 13, 14, 18.
  • Section 9.1 page 365:   1, 2, 4, 6, 7, 9,11 (do not use Euler's Theorem), 14, 17, 20 (Hint: You will need to use the Binomial Theorem as well as to prove that if p is prime then the binomial coefficient p!/k!(p-k)! is divisible by p, for 1< k < p).
  • Solution to 8.1.4, 8.2.16, 8.3.9, and 8.3.11.
• Assignment 6   (Due Thursday, March 26, in class, day of the midterm, material covered by the midterm)
  • Section 9.2 page 375:   3(b,d), 4(a, b), 7, 9, 19. Hint for 19: write m=p_1^{e_1}...p_k^{e_k}a, n=p_1^{f_1}...p_k^{f_k}b, where the exponets are positive, p_i are primes which do not divide neither a nor b, and gcd(a,b)=1..
  • Section 9.3 page 388:   1, 5, 15, 16 (a, b), 17a, 20.
  • Solution to 9.1.4, 9.1.9, 9.2.3, 9.3.15, 9.3.20.
• Assignment 7   (Due Thursday, April 2)
  • Section 9.5 page 408:   1, 3, 6, 8.
  • Section 10.1 page 425:   1, 2, 3(a), 5, 6, 7, 10, 11, 12, 14(a), 15(a).
  • Solution to 9.5.3, 9.5.8, 10.1.10, 10.1.15.
• Assignment 8   (Due Thursday, April 9)
  • Section 10.2 page 436:   2, 4, 7, 8, 9, 10, 13 (complete the square), 15 (Use the C.R.T. in part c).
  • Section 10.3 page 445:   3, 5, 6, 8, 9, 12, 13, 20 (you will need to use the Chinese Remainder Theorem in part c).
  • Solution to 10.2.8, 10.2.10, 10.3.8, 10.3.9.
• Assignment 9   (Due Thursday, April 16)
  • Section 11.1 page 475:   1, 3, 4, 5, 13, 14(a,b), 19, 26, 31 (Hint: There are two cases, depending on gcd(3,p-1)).
  • Section 11.2 page 488:   1(a,b), 2, 4, 10, 12.
  • Section 11.3 page 495:   1, 2, 5, 9, 10.
  • Solution to 11.1.4, 11.1.14, 11.2.2, 11.2.4, 11.3.9.
• Assignment 10   (Due Thursday, April 23)
  • Section 11.4 page 503:   1, 2, 7, 8, 9a, 11, 12, 13 (use the Chinese Remainder Theorem), 16 (Hint: show first that any solution x must be congruent to 3 or -3 mod 7), 20.
  • Solution to 11.4.8, 11.4.12, 11.4.13, 11.4.16.
  • Section 11.5 page 513:   7, 9, 12, 15, 22.
  • Solution to 11.5.22.
  • Section 11.6 page 523:   1, 4, 5, 6, 7.
  • Solution to 11.6.7.
• Assignment 11   (Due Thursday, April 30)
  • Section 14.1 page 660:   1, 4, 5, 8, 10, 12.
  • Section 14.2 page 660:   1, 5, 11.
• Assignment 12   (Due Thursday, May 7)
  • Section 14.2 page 660:   29, more to come...