Use the Euclidean algorithm to find the greatest common divisor of 412 and 32 and express it in terms of the two integers.

Step 1: Determine the largest integer that divides both numbers First, we will find the largest integer that can divide both 412 and 32 without leaving a remainder. We can do this using the Euclidean algorithm. The algorithm involves dividing the larger number by the smaller number and using the remainder as our new smaller number. We repeat this process until there is no remainder. The last non-zero remainder will be our GCD. Step 2: Set up the first division Divide the larger number (412) by the smaller number (32): $\frac{412}{32}=12\text{(remainder}28\text{)}$ Step 3: Repeat divisions with remainders as new smaller numbers Now, we'll use our remainder (28) to do further divisions until there's no remainder. 1. Divide the previous divisor (32) by the new remainder (28): $\frac{32}{28}=1\text{(remainder}4\text{)}$ 2. Divide the previous new remainder (28) by the new remainder (4): $\frac{28}{4}=7\text{(no remainder)}$ Since there's no remainder in this last step, the GCD of 412 and 32 is the last non-zero remainder: 4. Step 4: Express GCD as a linear combination of 412 and 32 Now, let's rewrite each division step as equation and solve for the remainder: 1. The first division: $412=12\times 32+28$ 2. The second division: $32=1\times 28+4$ From the second division equation, we have: $4=32-1\times 28\text{(1)}$ Now substitute the first division equation's expression for 28 into equation (1): $4=32-1\times (412-12\times 32)$ Distribute the -1 times the terms in the parenthesis and simplify: $4=-1\times 412+13\times 32$ Thus, the GCD(412,32) = 4 can be expressed as a linear combination of the two integers 412 and 32 as follows: $4=-1\times 412+13\times 32$