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Show that if n is odd and 3 n, then n2 = 1 (mod 24).

Show that if n is odd and 3 n, then n2 = 1 (mod 24).

Advanced Engineering Mathematics
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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Using Wilson's Theorm, Fermat's Little Theorm and The Pollard Factorization Metod how could I solve Quetion 18 in Section 6.1 

6.1 Wilson's Theorem and Fermat's Little Theorem 223 O Show that if n is odd and 3 n, then n2 = 1 (mod 24). 12 1 is divisible by 35 whenever (a, 35) = 1. 19. Show that a | 20 Show that a° - 1 is divisible by 168 whenever (a, 42) = 1. %3D 21. Show that 42 | (n' – n) for all positive integers n. 22. Show that 30 | (nº – n) for all positive integers n. 23. Show that 1P-1+2P-1+3P-1 has been conjectured that the converse of this is also true.) 1+. .+(p - 1)(P-1) = -1 (mod p) whenever p is prime. (It | 24. Show that 1P+ 2P + 3P + . + (p – 1)P = 0 (mod p) when p is an odd prime. 25. Show that if p is prime and a and b are integers not divisible by p, with aP = bP (mod p), then a = bP (mod p2). 26 Use the Pollard p - 1 method to find a divisor of 689. 27. Use the Pollard p – 1 method to find a divisor of 7,331,117. (For this exercise, you will need to use either a calculator or computational software.) - 28. Show that if p and q are distinct primes, then p9-1+ qP- =1 (mod pq). 29. Show that if p is prime and a is an integer, then p | (aP + (p – 1)!a). 30. Show that if p is an odd prime, then 1232 . .. (p – 4)2(p – 2)² = (-1) P+1)/2 (mod p). 31. Show that if p is prime and p =3 (mod 4), then ((p – 1)/2)!=±1 (mod p). 32. a) Let p be prime, and suppose that r is a positive integer less than p such that (-1)'r!= -1 (mod p). Show that (p – r + 1)! = -1 (mod p). b) Using part (a), show that 61!= 63!= -1 (mod 71). 33. Using Wilson's theorem, show that if p is a prime and p = 1 (mod 4), then the congruence x = -1 (mod p) has two incongruent solutions given by x = ±((p – 1)/2)! (mod p). 34. Show that if p is a prime and 0 < k < p, then (p – k)!(k – 1)!= (-1)* (mod p). 35. Show that if n is an integer, then 1 %3D j=2 20. Show that if p is a prime and p > 3, then 2P-2+ 3P-2+6P-2 = 1 (mod p). 37. Show that ifn is a nonnegative integer, then 5 | 1"+ 2" + 3" + 4" if and only if 4 n. +4" prime? * * 38. For which positive integers n is n tuin primes if and only if 4((n - 1)!+
Transcribed Image Text:6.1 Wilson's Theorem and Fermat's Little Theorem 223 O Show that if n is odd and 3 n, then n2 = 1 (mod 24). 12 1 is divisible by 35 whenever (a, 35) = 1. 19. Show that a | 20 Show that a° - 1 is divisible by 168 whenever (a, 42) = 1. %3D 21. Show that 42 | (n' – n) for all positive integers n. 22. Show that 30 | (nº – n) for all positive integers n. 23. Show that 1P-1+2P-1+3P-1 has been conjectured that the converse of this is also true.) 1+. .+(p - 1)(P-1) = -1 (mod p) whenever p is prime. (It | 24. Show that 1P+ 2P + 3P + . + (p – 1)P = 0 (mod p) when p is an odd prime. 25. Show that if p is prime and a and b are integers not divisible by p, with aP = bP (mod p), then a = bP (mod p2). 26 Use the Pollard p - 1 method to find a divisor of 689. 27. Use the Pollard p – 1 method to find a divisor of 7,331,117. (For this exercise, you will need to use either a calculator or computational software.) - 28. Show that if p and q are distinct primes, then p9-1+ qP- =1 (mod pq). 29. Show that if p is prime and a is an integer, then p | (aP + (p – 1)!a). 30. Show that if p is an odd prime, then 1232 . .. (p – 4)2(p – 2)² = (-1) P+1)/2 (mod p). 31. Show that if p is prime and p =3 (mod 4), then ((p – 1)/2)!=±1 (mod p). 32. a) Let p be prime, and suppose that r is a positive integer less than p such that (-1)'r!= -1 (mod p). Show that (p – r + 1)! = -1 (mod p). b) Using part (a), show that 61!= 63!= -1 (mod 71). 33. Using Wilson's theorem, show that if p is a prime and p = 1 (mod 4), then the congruence x = -1 (mod p) has two incongruent solutions given by x = ±((p – 1)/2)! (mod p). 34. Show that if p is a prime and 0 < k < p, then (p – k)!(k – 1)!= (-1)* (mod p). 35. Show that if n is an integer, then 1 %3D j=2 20. Show that if p is a prime and p > 3, then 2P-2+ 3P-2+6P-2 = 1 (mod p). 37. Show that ifn is a nonnegative integer, then 5 | 1"+ 2" + 3" + 4" if and only if 4 n. +4" prime? * * 38. For which positive integers n is n tuin primes if and only if 4((n - 1)!+
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