[Hint: It suffices to show that the numbe 11. Verify that 0, 1, 2, 2², 2³, . , 2° form a complete set of residues modulo 11, but that 0, 12, 22, 3², ., 10² do not. .... ..... toments:

Advanced Engineering Mathematics
10th Edition
ISBN:9780470458365
Author:Erwin Kreyszig
Publisher:Erwin Kreyszig
Chapter2: Second-order Linear Odes
Section: Chapter Questions
Problem 1RQ
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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 that
statements:
(-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 that
2n
= 0 (mod p)
10. If a1, a2, ..., a, is a complete set of residues modulo n and gcd(a, n) = 1, prove that
aaj, 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 that
0. 12, 2², 3². .., 10² do not.
12. Prove the following statements:
(a) If gcd(a, n) = 1, then the integers
с, с + а, с + 2а, с + За, ..., с + (п -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.
13. Verify that if a = b (mod n1) and a = b (mod n2), then a = b (mod n), where the integer
n = 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 > 1
j (mod n) need not imply that
a = 1 (mod 2"+2)
[Hint: Proceed by induction on n.]
16. Use the theory of congruences to verify that
89| 24 - 1
17, Prove that whenever ab = cd (mod n) and h=d (mad
and
97| 248 – 1
a =c (mod n)
then
Transcribed Image Text: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 that statements: (-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 that 2n = 0 (mod p) 10. If a1, a2, ..., a, is a complete set of residues modulo n and gcd(a, n) = 1, prove that aaj, 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 that 0. 12, 2², 3². .., 10² do not. 12. Prove the following statements: (a) If gcd(a, n) = 1, then the integers с, с + а, с + 2а, с + За, ..., с + (п -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. 13. Verify that if a = b (mod n1) and a = b (mod n2), then a = b (mod n), where the integer n = 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 > 1 j (mod n) need not imply that a = 1 (mod 2"+2) [Hint: Proceed by induction on n.] 16. Use the theory of congruences to verify that 89| 24 - 1 17, Prove that whenever ab = cd (mod n) and h=d (mad and 97| 248 – 1 a =c (mod n) then
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