1. 3*6 + -1*16 = 2, i.e., a=3, b=1.

You could calculate this using the extended Euclidean algorithm:
gcd(16,6) = gcd(6,4) = gcd(4,2) = gcd(2,0) = 2, so
    16 =  0*6 +  1*16     Eqn(1)
     6 =  1*6 +  0*16     Eqn(2)
     4 = -2*6 +  1*16     Eqn(3) = Eqn(1) - 2*Eqn(2)
     2 =  3*6 + -1*16     Eqn(4) = Eqn(2) - Eqn(3)

2. All the even integers, i.e., ...,-4,-2,0,2,4,6,...

Suppose n can be written as a linear combination of 6,16.  First, since
gcd(6,16) = 2, we know that 2 | n, i.e., n has to be even.  Since n is
even, it can be expressed as n=2k for some integer k.  Start from 3*6 +
-1*16 = 2, and multiply both sides by 2, to get (3*k)*6 + (-1*k)*16 =
2*k = n.  Therefore every even integer n can be expressed as a linear
combination of 6 and 16.