11 . Short and simplified. 18. If a =b (mod n1) and a =c (mod n2), prove that b =c (mod n), where the integer(a) 7|52n +3-2 5n –2(b) 13|3"+2 + 42n+1.(c) 27|25n+1 + 5n+2_(d) 43 6+2 +72n+1.7. For n > 1, show that4.3Onespechearto in(-13)"+1 = (-13)" + (-13)"- (mod 181)sysinte[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 a? = 1 (mod 8).(b) For any integer a, a³ = 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 thatth(:)-= 0 (mod p)10. If a1, a2, ..., an is a complete set of residues modulo n and gcd(a, n) = 1, prove thatad¡, aa2, . , aa, 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,..., 2º form a complete set of residues modulo 11, but that0, 12, 22, 3²,.., 10² do not.12. Prove the following statements:(a) If gcd(a, n) = 1, then the integersc, c+a, c+ 2a, c + 3a, ... , c + (n – 1)aform 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.13. Verify that if a = b (mod nj) and a = b (mod n2), then a = b (mod n), where the integern = lcm(n1, n2). Hence, whenever n¡ and n2 are relatively prime, a = b (mod n¡nɔ).14. Give an example to show that a = b* (mod n) and k = j (mod n) need not imply thatai = bi (mod n).15. Establish that if a is an odd integer, then for any n > 1a2"= 1 (mod 2"+2)[Hint: Proceed by induction on n.]16. Use the theory of congruences to verify that89| 244 197 | 248 1and17. Prove that whenever ab = cd (mod n) and b = d (mod n), with gcd(b, n) = 1. thena =c (mod n).n = gcd(n1, n2).

Name: 11 . Short and simplified. 18. If a =b (mod n1) and a =c (mod n2), pr
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Description: 11 . Short and simplified. 18. If a =b (mod n1) and a =c (mod n2), prove that b =c (mod n), where the integer(a) 7|52n +3-2 5n –2(b) 13|3"+2 + 42n+1.(c) 27|25n+1 + 5n+2_(d) 43 6+2 +72n+1.7. For n > 1, show that4.3Onespechearto in(-13)"+1 = (-13)" +