Question: b) Given $a = 2020$ and $b = 92$. (i) Use the Euclidean algorithm to find the Greatest Common Divisor (GCD) of $a$ and $b$. (ii) Present the Greatest Common Divisor (GCD) $a$ and $b$ in the form of $sa + tb$ where $s, t \in \mathbb{Z}$. (iii) Hence, compute the Least Common Multiple (LCM) of $a$ and $b$.

1. **State the problem:** We are given $a = 2020$ and $b = 92$. (i) Find the GCD of $a$ and $b$ using the Euclidean algorithm. (ii) Express the GCD as $sa + tb$ for some integers $s$ and $t$. (iii) Compute the LCM of $a$ and $b$. --- 2. **Euclidean Algorithm to find GCD:** The Euclidean algorithm uses repeated division: $$\text{GCD}(a,b) = \text{GCD}(b, a \bmod b)$$ Calculate step-by-step: $$2020 = 92 \times 21 + 88$$ $$92 = 88 \times 1 + 4$$ $$88 = 4 \times 22 + 0$$ Since remainder is $0$ at last step, the GCD is the last non-zero remainder: $$\boxed{\text{GCD}(2020, 92) = 4}$$ --- 3. **Express GCD as $sa + tb$ (Extended Euclidean Algorithm):** Start from the second last step: $$4 = 92 - 88 \times 1$$ Substitute $88$ from the first step: $$88 = 2020 - 92 \times 21$$ So, $$4 = 92 - (2020 - 92 \times 21) \times 1 = 92 - 2020 + 92 \times 21 = 92 \times 22 - 2020$$ Rewrite: $$4 = (-1) \times 2020 + 22 \times 92$$ Thus, $$s = -1, \quad t = 22$$ So, $$\boxed{4 = (-1) \times 2020 + 22 \times 92}$$ --- 4. **Compute LCM:** Recall the relation: $$\text{LCM}(a,b) = \frac{|a \times b|}{\text{GCD}(a,b)}$$ Calculate: $$\text{LCM}(2020, 92) = \frac{2020 \times 92}{4}$$ Simplify by canceling common factor 4: $$= 2020 \times \cancel{\frac{92}{4}} = 2020 \times 23 = 46460$$ So, $$\boxed{\text{LCM}(2020, 92) = 46460}$$