Problem 1TFE: True or False
Label each of the following statements as either true or false.
1. implies for .
Problem 2TFE: True or False
Label each of the following statements as either true or false.
2. and imply for... Problem 3TFE: Label each of the following statements as either true or false. a2b2(modn) and implies ab(modn) or... Problem 4TFE: Label each of the following statements as either true or false. a is congruent to b modulo n if and... Problem 5TFE: Label each of the following statements as either true or false. The distinct congruence classes for... Problem 6TFE: Label each of the following statements as either true or false. If ab0(modn), then either a0(modn)... Problem 7TFE: Label each of the following statements as either true or false. If (a,n)=1, then a1(modn). Problem 1E: In this exercise set, all variables are integers.
1. List the distinct congruence classes modulo ,... Problem 2E: In this exercise set, all variables are integers.
2. Follow the instructions in Exercise for the... Problem 3E: Find a solution , , for each of the congruences in Exercises.
Note that in each case, and are... Problem 4E: Find a solution , , for each of the congruences in Exercises.
Note that in each case, and are... Problem 5E: Find a solution x, 0xn, for each of the congruences axb(modn) in Exercises 324. Note that in each... Problem 6E Problem 7E: Find a solution x, 0xn, for each of the congruences axb(modn) in Exercises 324. Note that in each... Problem 8E: Find a solution x, 0xn, for each of the congruences axb(modn) in Exercises 324. Note that in each... Problem 9E: Find a solution , , for each of the congruences in Exercises.
Note that in each case, and are... Problem 10E Problem 11E: Find a solution , , for each of the congruences in Exercises.
Note that in each case, and are... Problem 12E Problem 13E: Find a solution x, 0xn, for each of the congruences axb(modn) in Exercises 324. Note that in each... Problem 14E Problem 15E: Find a solution x, 0xn, for each of the congruences axb(modn) in Exercises 324. Note that in each... Problem 16E Problem 17E: Find a solution x, 0xn, for each of the congruences axb(modn) in Exercises 324. Note that in each... Problem 18E: Find a solution x, 0xn, for each of the congruences axb(modn) in Exercises 324. Note that in each... Problem 19E: Find a solution x, 0xn, for each of the congruences axb(modn) in Exercises 324. Note that in each... Problem 20E Problem 21E Problem 22E Problem 23E Problem 24E: Find a solution , , for each of the congruences in Exercises.
Note that in each case, and are... Problem 25E: 25. Complete the proof of Theorem : If and is any integer, then
.
Problem 26E: Complete the proof of Theorem 2.24: If ab(modn) and cd(modn), then a+cb+d(modn). Problem 27E: Prove that if a+xa+y(modn), then xy(modn). Problem 28E: 28. If and where , prove that .
Problem 29E: 29. Find the least positive integer that is congruent to the given sum, product, or power.
a. ... Problem 30E: 30. Prove that any positive integer is congruent to its units digit modulo .
Problem 31E: 31. If , prove that for every positive integer .
Problem 32E: 32. Prove that if is an integer, then either or . (Hint: Consider the cases where is even and... Problem 33E: Prove or disprove that if n is odd, then n21(mod8). Problem 34E Problem 35E Problem 36E Problem 37E Problem 38E Problem 39E Problem 40E: In the congruences axb(modn) in Exercises 4053, a and n may not be relatively prime. Use the results... Problem 41E: In the congruences in Exercises, and may not be relatively prime. Use the results in Exercises and ... Problem 42E: In the congruences in Exercises, and may not be relatively prime. Use the results in Exercises and ... Problem 43E: In the congruences axb(modn) in Exercises 4053, a and n may not be relatively prime. Use the results... Problem 44E: In the congruences in Exercises, and may not be relatively prime. Use the results in Exercises and ... Problem 45E Problem 46E: In the congruences in Exercises, and may not be relatively prime. Use the results in Exercises and ... Problem 47E Problem 48E Problem 49E: In the congruences in Exercises, and may not be relatively prime. Use the results in Exercises and ... Problem 50E: In the congruences in Exercises, and may not be relatively prime. Use the results in Exercises and ... Problem 51E: In the congruences ax b (mod n) in Exercises 40-53, a and n may not be relatively prime. Use the... Problem 52E: In the congruences axb(modn) in Exercises 4053, a and n may not be relatively prime. Use the results... Problem 53E Problem 54E: 54. Let be a prime integer. Prove Fermat's Little Theorem: For any positive integer,. (Hint: Use... Problem 55E: 55. Prove the Chinese Remainder Theorem: Let , , . . . , be pairwise relatively prime. There exists... Problem 56E: 56. Solve the following systems of congruences.
a. b.
c. d.
e.... Problem 57E Problem 58E: a. Prove that 10n(1)n(mod11) for every positive integer n. b. Prove that a positive integer z is... format_list_bulleted