(a)Use the Euclidean algorithm as follows:
78 = 35.2 + 8
35 = 8.4 + 3
8 = 3.2 + 2
3 = 2.1 + 1
2 = 2.1
The last non zero remainder is 1 so the gcd(35,78) is 1. To find the requisite linear combination work backwards from the second last step as follows:
1 = 3 - 2.1 = 3 - (8-3.2) = 3 - 8 + 3.2 = 3.3 - 8 = 3.(35 - 8.4) - 8 = 3.35 - 8.12 - 8 = 3.35 - 8.13 = 3.35 - 13.(78-35.2) = 3.35 - 13.78 + 35.26 = 29.35 - 13.78
Notice that I never actually multiplied out and added the terms. I only brought each equation in a form so as to allow me to substitute the previous step's remainder in it.
Similarly,
(b) gcd(101,203) = 1 = -2.101 + 1.203
(c) gcd(2002,2339) = 1 = -819.2002 + 701.2339
To find the inverse of 2 modulo 17 first note that the inverse exists iff gcd(2,17) = 1 which is true here. Now apply the Euclidean algorithm as in steps (a), (b) and (c) to obtain 1 = -8.2 + 1.17 or in other words 1 - (-8).2 = 17. Thus -8.2 = 1 mod 17 or -8 is the inverse of 2 mod 17. Obviously -8 = 9 mod 17 so the inverse is 9.