6a ENTARY NUMBER THEORY14. Give an example to show that ak = b* (mod n) and k = j (mod n) need not imply thatn = lcm(n1, n2). Hence, whenever n1 and n2 are relatively prime, a = b (mod n¡n2).0. For n > 1, use congruence theory to establish each of the following divisibilitystatements:(a) 7|52n +3. 25n-2.(b) 13| 3"+2 + 42n+1,(c) 27|25n+1 + 5"+2.(d) 43 6"+2 + 72n+17. For n > 1, show that(-13)"+1 = (-13)" + (-13)"-1 (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, 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 that2n= 0 (mod p)10. If a1, a2, ..., a, is a complete set of residues modulo n and gcd(a, n) = 1, prove thataa1, aa2, . , aa, is also a complete set[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, 32,..., 10² do not.12. Prove the following statements:(a) If gcd(a, n) = 1, then the integersresidues modulo n.2° form a complete set of residues modulo 11, but thatс, с +а, с + 2а, с + За, ..., с + (n - 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 n¡) and a = b (mod nɔ), then a = b (mod n), where the mesai = bi (mod n).15. Establish that if a is an odd integer, then for any n 2 1a2" = 1 (mod 2"+2)[Hint: Proceed by induction on n.]16. Use the theory of

Name: 6a ENTARY NUMBER THEORY14. Give an example to show that ak = b* (mod n
Uploaded: 2024-03-20T07:06:41.000Z
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Description: 6a ENTARY NUMBER THEORY14. Give an example to show that ak = b* (mod n) and k = j (mod n) need not imply thatn = lcm(n1, n2). Hence, whenever n1 and n2 are relatively prime, a = b (mod n¡n2).0. For n > 1, use congruence theory to establish each of th