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B. Prove the assertions below: (a) If a is an odd integer, then a? = 1 (mod 8). (b) For any integer a, a = 0, 1, or 6 (mod 7).

B. Prove the assertions below: (a) If a is an odd integer, then a? = 1 (mod 8). (b) For any integer a, a = 0, 1, or 6 (mod 7).

Elements Of Modern Algebra
Elements Of Modern Algebra
8th Edition
ISBN:9781285463230
Author:Gilbert, Linda, Jimmie
Publisher:Gilbert, Linda, Jimmie
Chapter2: The Integers
Section2.5: Congruence Of Integers
Problem 58E: a. Prove that 10n(1)n(mod11) for every positive integer n. b. Prove that a positive integer z is...
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6. For n 1, use congruence theory to establish each of the following divisibili 13. Verify that if a = b (modn¡) and a = b (mod n2), then a = b'(mod n), where the integer n = lcm(n1, n2). Hence, whenever nj and n, are relatively prime, a = b (mod n¡n2). statements: (a) 7|52n +3- 25n–2. (b) 13 3"+2 + 42n+1. (c) 27|25n+1 + 5n+2. (d) 43|6"+2 + 72n+1. 7. For n > 1, show that (-13)"+1 = (-13)" +(-13)"-' (mod 181) [Hint: Notice that (-13)2 = -13+1 (mod 181); use induction on n.] 8. Prove the assertions below: (a) If a is an odd integer, then a2 = 1 (mod 8). (b) For any integer a, a' = 0, 1, or 6 (mod 7). (c) For any integer a, a* = 0 or 1 (mod 5). (d) If the integer a is not divisible by 2 or 3, then a2 = 1 (mod 24). 9. If p is a prime satisfying n < p < 2n, show that (:)- 2n = 0 (mod p) n 10. If a1, a2, . .., an is a complete set of residues modulon and gcd(a, n) = 1, prove tha aaj, aa2, ... , aan is also a complete set of residues modulo n. [Hint: It suffices to show that the numbers in question are incongruent modulo n.] 11. Verify that 0, 1, 2, 2², 2³, 0, 12, 22, 3?, . 12. Prove the following statements: (a) If gcd(a, n) = 1, then the integers 2° form a complete set of residues modulo 11, but that 102 do not. с, с +а, с +2а, с + За, ..., с+ (n - 1)а form a complete set of residues modulo n for any c. (b) Any n consecutive integers form a complete set of residues modulo n. [Hint: Use part (a).] (c) The product of any set of n consecutive integers is divisible by n. 14. Give an example to show that ak = bk (mod n) and k = j (mod n) need not imply ai = bi (mod n). 15. Establish that if a is an odd integer, then for any n> 1 that a" = 1 (mod 2"+2) (Hint: Procaod l
Transcribed Image Text:6. For n 1, use congruence theory to establish each of the following divisibili 13. Verify that if a = b (modn¡) and a = b (mod n2), then a = b'(mod n), where the integer n = lcm(n1, n2). Hence, whenever nj and n, are relatively prime, a = b (mod n¡n2). statements: (a) 7|52n +3- 25n–2. (b) 13 3"+2 + 42n+1. (c) 27|25n+1 + 5n+2. (d) 43|6"+2 + 72n+1. 7. For n > 1, show that (-13)"+1 = (-13)" +(-13)"-' (mod 181) [Hint: Notice that (-13)2 = -13+1 (mod 181); use induction on n.] 8. Prove the assertions below: (a) If a is an odd integer, then a2 = 1 (mod 8). (b) For any integer a, a' = 0, 1, or 6 (mod 7). (c) For any integer a, a* = 0 or 1 (mod 5). (d) If the integer a is not divisible by 2 or 3, then a2 = 1 (mod 24). 9. If p is a prime satisfying n < p < 2n, show that (:)- 2n = 0 (mod p) n 10. If a1, a2, . .., an is a complete set of residues modulon and gcd(a, n) = 1, prove tha aaj, aa2, ... , aan is also a complete set of residues modulo n. [Hint: It suffices to show that the numbers in question are incongruent modulo n.] 11. Verify that 0, 1, 2, 2², 2³, 0, 12, 22, 3?, . 12. Prove the following statements: (a) If gcd(a, n) = 1, then the integers 2° form a complete set of residues modulo 11, but that 102 do not. с, с +а, с +2а, с + За, ..., с+ (n - 1)а form a complete set of residues modulo n for any c. (b) Any n consecutive integers form a complete set of residues modulo n. [Hint: Use part (a).] (c) The product of any set of n consecutive integers is divisible by n. 14. Give an example to show that ak = bk (mod n) and k = j (mod n) need not imply ai = bi (mod n). 15. Establish that if a is an odd integer, then for any n> 1 that a" = 1 (mod 2"+2) (Hint: Procaod l
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