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 Theorem223O Show that if n is odd and 3 n, then n2 = 1 (mod 24).121 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.%3D21. 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-1has 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 needto 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 congruencex = -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

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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 Theorem223O Show that if n is odd and 3 n, then n2 = 1 (mod 24).121 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.%3D21. 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-1has 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 needto 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 congruencex = -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, then1%3Dj=220. 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&#34;+ 2&#34; + 3&#34; + 4&#34; if and only if 4 n.+4&#34; prime?** 38. For which positive integers n is ntuin primes if and only if 4((n - 1)!+

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The solution is below.