Can you do #5 Julowe see that10. Write a computer program that determines if an element a is invertible mod-to the congruence 3x = 7 (mod 17) is x = 8 (mod 17). Then the7. If m is composite, what is the value (m – 1)! mod m in the standard residue6.3-1.17 = 1, so the inverse of 3 modulo 17 is 6. Hence the solution119/7= 17. Applying the extended Euclidean Algorithm.solutions to 21x = 49 (mod 119) are 8, 25, 42, 59, 76, 93, and 110Exercises for Section 3.21. Determine the invertible elements modulo 15, 17, and 32.2. Determine the inverse of 67 modulo 119.3. Determine all solutions to the congruences 7x = 5 (mod 13) and 4r =6(mod 18).4. Solve the following linear congruences.(a) 11x = 28 (mod 37).(b) 42x = 90 (mod 156).J 5. If a-l is the inverse of a modulo m and b-1 the inverse of b modulo m,show that a- is the inverse of ab modulo m.6. Let p be an odd prime.(a) What are the possible values of x mod p so that x is its own inversemodulo p?(b) Consider the product 1 · 2 . . (p – 1) = (p – 1)! (mod p). Showpairing the elements with their inverses that (p – 1)! = -1 (mod P.This result is known as Wilson's Theorem. (Does this match wiuiyour conjecture in Exercise 3.1.23?)system? Conclude that (m – 1)! = -1 (mod m) implies that m is pin8. Use Exercise 6 to answer the following questions:(a) Determine 65! mod 67.(b) If p is a prime number, what is (p – 2)! mod p?9. Show that if p = 4k + 1 is prime, then [(2k)!]² := -1 (mod p).ulo m, and if it is, the program computes the inverse.11. Write a computer program to solve linear congruences modulo t.

Name: Can you do #5 Julowe see that10. Write a computer program that determi
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Description: Can you do #5 Julowe see that10. Write a computer program that determines if an element a is invertible mod-to the congruence 3x = 7 (mod 17) is x = 8 (mod 17). Then the7. If m is composite, what is the value (m – 1)! mod m in the standard residue6.3