Use the Extended Euclidean algorithm to find the GCD of 20 and 24.

The Extended Euclidean Algorithm is used to find the greatest common divisor (GCD) of two integers, along with coefficients (x and y) such that $ax+by=\text{GCD}(a,b)$. Here, we apply it to find the GCD of 16 and 28.

Step-by-Step Solution:

1. Initialization: We start with $a=28$ and $b=16$. The algorithm proceeds by repeatedly applying the division algorithm: $$a=b\cdot q+r\text{where}0\le r<b$$

2. First Iteration: Divide 28 by 16: $$28=16\times 1+12\text{(Remainder }r=12\text{)}$$ Now, replace $a$ with $b$ and $b$ with $r$: $$\text{GCD}(28,16)=\text{GCD}(16,12)$$

3. Second Iteration: Divide 16 by 12: $$16=12\times 1+4\text{(Remainder }r=4\text{)}$$ Update $a$ and $b$: $$\text{GCD}(16,12)=\text{GCD}(12,4)$$

4. Third Iteration: Divide 12 by 4: $$12=4\times 3+0\text{(Remainder }r=0\text{)}$$ Since the remainder is now 0, the algorithm terminates. The last non-zero remainder is the GCD: $$\text{GCD}(12,4)=4$$

Final Result:

The GCD of 16 and 28 is 4.

Finding Coefficients (Optional):

To find integers $x$ and $y$ such that $16x+28y=4$, we back-substitute the equations:

1. From Step 2: $$12=28-16\times 1$$
2. From Step 3: $$4=16-12\times 1$$ Substitute $12$ from Step 2: $$4=16-(28-16\times 1)\times 1=2\times 16-1\times 28$$ Thus, $x=2$ and $y=-1$.

Verification:

$$16\times 2+28\times (-1)=32-28=4$$ This confirms the correctness of the solution.