Use the Euclidean algorithm to hand-calculate the greatest common divisor for the integers given below.
479 and 1,469
Step 1: Find q1 and r1 so that
1,469 = 479 · q1 + r1,   where 0 ≤ r1 < 479.
Then 
32
 = 1,469 − 479 · q1.
Step 2: Find q2 and r2 so that
479 = r1 · q2 + r2,   where 0 ≤ r2 < r1.
Then 
 = 479 − 

 
 

 · q2.
Step 3: Find q3 and r3 so that
r1 = r2 · q3 + r3,   where 0 ≤ r3 < r2.
Then 
 = 
 − 

 
 

 · q3.
Step 4: Find q4 and r4 so that
r2 = r3 · q4 + r4,   where 0 ≤ r4 < r3.
Then 
 = 
 − 

 
 

 · q4.
Step 5: Conclude that gcd(1,469, 479) = 
 .


To find the greatest common divisor (gcd) of 1,469 and 479 using the Euclidean algorithm, let's follow the steps provided. We want to apply the division algorithm repeatedly until we reach a remainder of 0.

### Step 1:
We begin by dividing 1,469 by 479.

\[
1,469 = 479 \cdot q_1 + r_1
\]

Perform the division:

\[
1,469 \div 479 = 3 \text{ (quotient)}
\]

Now calculate the remainder \(r_1\):

\[
1,469 - 479 \cdot 3 = 1,469 - 1,437 = 32
\]

Thus, we have:

\[
1,469 = 479 \cdot 3 + 32
\]

So, \(q_1 = 3\) and \(r_1 = 32\).

### Step 2:
Next, we divide 479 by 32.

\[
479 = 32 \cdot q_2 + r_2
\]

Perform the division:

\[
479 \div 32 = 14 \text{ (quotient)}
\]

Now calculate the remainder \(r_2\):

\[
479 - 32 \cdot 14 = 479 - 448 = 31
\]

Thus, we have:

\[
479 = 32 \cdot 14 + 31
\]

So, \(q_2 = 14\) and \(r_2 = 31\).

### Step 3:
Now, divide 32 by 31.

\[
32 = 31 \cdot q_3 + r_3
\]

Perform the division:

\[
32 \div 31 = 1 \text{ (quotient)}
\]

Now calculate the remainder \(r_3\):

\[
32 - 31 \cdot 1 = 32 - 31 = 1
\]

Thus, we have:

\[
32 = 31 \cdot 1 + 1
\]

So, \(q_3 = 1\) and \(r_3 = 1\).

### Step 4:
Now, divide 31 by 1.

\[
31 = 1 \cdot q_4 + r_4
\]

Perform the division:

\[
31 \div 1 = 31 \text{ (quotient)}
\]

Now calculate the remainder \(r_4\):

\[
31 - 1 \cdot 31 = 31 - 31 = 0
\]

Thus, we have:

\[
31 = 1 \cdot 31 + 0
\]

So, \(q_4 = 31\) and \(r_4 = 0\).

### Conclusion:
Since the remainder is now 0, the gcd is the last non-zero remainder, which is 1.

Therefore, the gcd(1,469, 479) = **1**.

Would you like more details on any of these steps or another related topic?

Here are 5 related questions to expand your understanding:
1. How does the Euclidean algorithm work for larger numbers?
2. What is the difference between gcd and lcm (least common multiple)?
3. Can the Euclidean algorithm be used for more than two numbers?
4. How can the extended Euclidean algorithm find linear combinations?
5. What are practical applications of the gcd in number theory?

**Tip:** The Euclidean algorithm is very efficient and works even for large numbers, with only a few iterations required to find the gcd.

Use the Euclidean algorithm to hand-calculate the greatest common divisor for the integers given below. 479 and 1,469. Step 1: Find q1 and r1 so that 1,469 = 479 · q1 + r1, where 0 ≤ r1 < 479. Then 32 = 1,469 − 479 · q1. Step 2: Find q2 and r2 so that 479 = r1 · q2 + r2, where 0 ≤ r2 < r1. Then  = 479 −  · q2. Step 3: Find q3 and r3 so that r1 = r2 · q3 + r3, where 0 ≤ r3 < r2. Then  = − · q3. Step 4: Find q4 and r4 so that r2 = r3 · q4 + r4, where 0 ≤ r4 < r3. Then  = − · q4. Step 5: Conclude that gcd(1,469, 479) = .

Learn how to calculate the greatest common divisor (GCD) of 1,469 and 479 using the Euclidean algorithm. This detailed step-by-step guide covers the division process, quotients, and remainders at each step, leading to the final GCD result. Suitable for high school students studying number theory.

Math Problem Statement

· q2. Step 3: Find q3 and r3 so that r1 = r2 · q3 + r3, where 0 ≤ r3 < r2. Then

· q3. Step 4: Find q4 and r4 so that r2 = r3 · q4 + r4, where 0 ≤ r4 < r3. Then

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