11 6. For n(a) 7|52 + 3. 25n-2(b) 13|3"+2 + 42n+1.(c) 2712Sn+1 + 5n+2.(d) 4316"+2 + 72n+1.7. For n > 1, show thatstatements:(-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 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 a? = 1 (mod 24).9. If p is a prime satisfying n < p < 2n, show that2n= 0 (mod p)10. If a1, a2, ..., a, is a complete set of residues modulo n and gcd(a, n) = 1, prove thataaj, 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, . ., 2° 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 integersс, с + а, с + 2а, с + За, ..., с + (п -1)аform 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 n1) and a = b (mod n2), then a = b (mod n), where the integern = lcm(n1, n2). Hence, wheneverni and n2 are relatively prime, a = b (mod n1n2).14. Give an example to show that ak = bk (mod n) and k =ai = bi (mod n).15. Establish that if a is an odd integer, then for any n > 1j (mod n) need not imply thata = 1 (mod 2"+2)[Hint: Proceed by induction on n.]16. Use the theory of congruences to verify that89| 24 - 117, Prove that whenever ab = cd (mod n) and h=d (madand97| 248 – 1a =c (mod n)then

Name: 11 6. For n(a) 7|52 + 3. 25n-2(b) 13|3"+2 + 42n+1.(c) 2712Sn+1 + 5n+2.
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Description: 11 6. For n(a) 7|52 + 3. 25n-2(b) 13|3"+2 + 42n+1.(c) 2712Sn+1 + 5n+2.(d) 4316"+2 + 72n+1.7. For n > 1, show thatstatements:(-13)"+1 = (-13)" + (-13)"- (mod 181)[Hint: Notice that (–13)² = –13+1 (mod 181); use induction on n.]8. Prove the assertions