B. 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).
B. 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).
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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Equations and Inequations
Equations and inequalities describe the relationship between two mathematical expressions.
Linear Functions
A linear function can just be a constant, or it can be the constant multiplied with the variable like x or y. If the variables are of the form, x2, x1/2 or y2 it is not linear. The exponent over the variables should always be 1.
Question
8a,b
![6. For n 1, use congruence theory to establish each of the following divisibili
13. Verify that if a = b (modn¡) and a = b (mod n2), then a = b'(mod n), where the integer
n = lcm(n1, n2). Hence, whenever nj and n, are relatively prime, a = b (mod n¡n2).
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)"-' (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, a' = 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)
n
10. If a1, a2, . .., an is a complete set of residues modulon and gcd(a, n) = 1, prove tha
aaj, aa2, ... , aan 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,
0, 12, 22, 3?, .
12. Prove the following statements:
(a) If gcd(a, n) = 1, then the integers
2° form a complete set of residues modulo 11, but that
102 do not.
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.
14. Give an example to show that ak = bk (mod n) and k = j (mod n) need not imply
ai = bi (mod n).
15. Establish that if a is an odd integer, then for any n> 1
that
a" = 1 (mod 2"+2)
(Hint: Procaod l](/v2/_next/image?url=https%3A%2F%2Fcontent.bartleby.com%2Fqna-images%2Fquestion%2F7dbb4ae4-0d65-4baa-9481-63f79be91eca%2F90785cd7-7b26-4abb-92ea-ad0262652b66%2F87iql2m_processed.jpeg&w=3840&q=75)
Transcribed Image Text:6. For n 1, use congruence theory to establish each of the following divisibili
13. Verify that if a = b (modn¡) and a = b (mod n2), then a = b'(mod n), where the integer
n = lcm(n1, n2). Hence, whenever nj and n, are relatively prime, a = b (mod n¡n2).
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)"-' (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, a' = 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)
n
10. If a1, a2, . .., an is a complete set of residues modulon and gcd(a, n) = 1, prove tha
aaj, aa2, ... , aan 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,
0, 12, 22, 3?, .
12. Prove the following statements:
(a) If gcd(a, n) = 1, then the integers
2° form a complete set of residues modulo 11, but that
102 do not.
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.
14. Give an example to show that ak = bk (mod n) and k = j (mod n) need not imply
ai = bi (mod n).
15. Establish that if a is an odd integer, then for any n> 1
that
a" = 1 (mod 2"+2)
(Hint: Procaod l
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