By using the Euclidean Algorithm determine:
(a) $\gcd (156, 18)$
(b) $\gcd (129, 1011)$
(c) $\gcd (703, 111)$
(d) $\gcd (181, 232)$
(a) Applying the Division Algorithm to 156 and 18 gives
$$ 156 = 8 (18) + 12 $$
We apply the Division Algorithm to 18 and the non-zero remainder 12.
$$ 18 = 1(12) + 6 $$
We apply the Division Algorithm to 12 and the non-zero remainder 6.
$$ 12 = 2(6) + 0 $$
The algorithm terminates. The last non-zero remainder is 6, so $\gcd(156,18)=6$.
(b) Applying the Division Algorithm to 1011 and 129 gives
$$ 1011 = 7(129) + 108 $$
We apply the Division Algorithm to 129 and 108.
$$ 129 = 1(108) + 21 $$
We apply the Division Algorithm to 108 and 21.
$$ 108 = 5(21) + 3 $$
We apply the Division Algorithm to 21 and 3.
$$ 21 = 7(3) + 0 $$
The last non-zero remainder is 3, so $\gcd(1011,129) = 3$
(c) We apply the Division Algorithm to 703 and 111.
$$ 703 = 6(111) + 37$$
$$ 111 = 3(37) + 0 $$
So $\gcd(703,111) = 37$
(d) We apply the Division Algorithm to 232 and 181.
$$ 232 = 1(181) + 51 $$
$$ 181 = 3(51) + 28 $$
$$ 51 = 1(28) + 23 $$
$$ 28 = 1(23) + 5 $$
$$ 23 = 4(5) + 3 $$
$$ 5 = 1(3) + 2 $$
$$ 3 = 1(2) + 1 $$
$$ 2 = 1(2) + 0 $$
So $\gcd(232,181) = 1$. They are coprime.