4.13.  The non-zero elements of Z/5Z are [1], [2], [3], and [4].  Their
    inverses are

x  | x^-1
---------
1  | 1
2  | 3
3  | 2
4  | 4

4.15. The vectors computed during the execution of the extended
      Euclidean algorithm are given below

a.

[1, 0, 24140]
[0, 1, 16762]
[1, -1, 7378]
[-2, 3, 2006]
[7, -10, 1360]
[-9, 13, 646]
[25, -36, 68]
[-234, 337, 34]
[493, -710, 0]

So gcd(24140,16762) = 34

b. 

[1, 0, 12075]
[0, 1, 4655]
[1, -2, 2765]
[-1, 3, 1890]
[2, -5, 875]
[-5, 13, 140]
[32, -83, 35]
[-133, 345, 0]

So gcd(12075, 4655) = 35

4.19.  Again, the extended Euclidean algorithm output is:

a. 
[1, 0, 4321]
[0, 1, 1234]
[1, -3, 619]
[-1, 4, 615]
[2, -7, 4]
[-307, 1075, 3]
[309, -1082, 1]
[-1234, 4321, 0]

So 1234^-1 = -1082 = 3239 mod 4321

b. 

[1, 0, 40902]
[0, 1, 24140]
[1, -1, 16762]
[-1, 2, 7378]
[3, -5, 2006]
[-10, 17, 1360]
[13, -22, 646]
[-36, 61, 68]
[337, -571, 34]
[-710, 1203, 0]

Since gcd(40902,24140)=34 != 1, 24140 does not have an inverse mod 40902.

c.

[1, 0, 1769]
[0, 1, 550]
[1, -3, 119]
[-4, 13, 74]
[5, -16, 45]
[-9, 29, 29]
[14, -45, 16]
[-23, 74, 13]
[37, -119, 3]
[-171, 550, 1]
[550, -1769, 0]

So 550^-1 = 550 mod 1769.