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.