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If p is a prime satisfying n < p < 2n, show that 2n = 0 (mod p)

If p is a prime satisfying n < p < 2n, show that 2n = 0 (mod p)

Elements Of Modern Algebra
Elements Of Modern Algebra
8th Edition
ISBN:9781285463230
Author:Gilbert, Linda, Jimmie
Publisher:Gilbert, Linda, Jimmie
Chapter2: The Integers
Section2.7: Introduction To Coding Theory (optional)
Problem 12E: Suppose that the check digit is computed as described in Example . Prove that transposition errors...
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Download 2 Open v E Workflows v 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 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, ..., an is a complete set of residues modulo n and gcd(a aaj, aa2, . . , aaŋ is also a complete set of residues modulo n. [Hint: It suffices to show that the numbers in question are incongru 11. Verify that 0, 1, 2, 22, 2', ..., 2° form a complete set of residues 0, 12, 22, 32,..., 102 do not. 12. Prove the following statements: (a) If gcd(a, n) = 1, then the integers 1la
Transcribed Image Text:Download 2 Open v E Workflows v 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 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, ..., an is a complete set of residues modulo n and gcd(a aaj, aa2, . . , aaŋ is also a complete set of residues modulo n. [Hint: It suffices to show that the numbers in question are incongru 11. Verify that 0, 1, 2, 22, 2', ..., 2° form a complete set of residues 0, 12, 22, 32,..., 102 do not. 12. Prove the following statements: (a) If gcd(a, n) = 1, then the integers 1la
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