LLEMENTARY NUMBER THEORY 0. For n 2 1, use congruence theory to establish each of the following divisibility 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)"-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 a? = 1 (mod 8). (b) For any integer a, a = 0, 1, or 6 (mod 7). (c) For
LLEMENTARY NUMBER THEORY 0. For n 2 1, use congruence theory to establish each of the following divisibility 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)"-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 a? = 1 (mod 8). (b) For any integer a, a = 0, 1, or 6 (mod 7). (c) For
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Binomial is an algebraic expression of the sum or the difference of two terms. Before knowing about binomial distribution, we must know about the binomial theorem.
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6a
![ENTARY NUMBER THEORY
14. Give an example to show that ak = b* (mod n) and k = j (mod n) need not imply that
n = 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 divisibility
statements:
(a) 7|52n +3. 25n-2.
(b) 13| 3"+2 + 42n+1,
(c) 27|25n+1 + 5"+2.
(d) 43 6"+2 + 72n+1
7. 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 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
aa1, 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 integers
residues modulo n.
2° form a complete set of residues modulo 11, but that
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.
13. Verify that-if a = b (mod n¡) and a = b (mod nɔ), then a = b (mod n), where the mes
ai = bi (mod n).
15. Establish that if a is an odd integer, then for any n 2 1
a2" = 1 (mod 2"+2)
[Hint: Proceed by induction on n.]
16. Use the theory of](/v2/_next/image?url=https%3A%2F%2Fcontent.bartleby.com%2Fqna-images%2Fquestion%2F7dbb4ae4-0d65-4baa-9481-63f79be91eca%2Febe5354b-c9da-406c-93aa-4657d0cdbb67%2Fctffi5s_processed.jpeg&w=3840&q=75)
Transcribed Image Text:ENTARY NUMBER THEORY
14. Give an example to show that ak = b* (mod n) and k = j (mod n) need not imply that
n = 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 divisibility
statements:
(a) 7|52n +3. 25n-2.
(b) 13| 3"+2 + 42n+1,
(c) 27|25n+1 + 5"+2.
(d) 43 6"+2 + 72n+1
7. 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 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
aa1, 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 integers
residues modulo n.
2° form a complete set of residues modulo 11, but that
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.
13. Verify that-if a = b (mod n¡) and a = b (mod nɔ), then a = b (mod n), where the mes
ai = bi (mod n).
15. Establish that if a is an odd integer, then for any n 2 1
a2" = 1 (mod 2"+2)
[Hint: Proceed by induction on n.]
16. Use the theory of
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