<Question>
Find the greatest common divisor of $3339$, $2961$, and $1491$.
</Question>

<think>
<Plan_1>
Okay, so I need to find the greatest common divisor (GCD) of three numbers: 3339, 2961, and 1491. Hmm, let me remember how to do this. The GCD of multiple numbers is the largest number that divides all of them without leaving a remainder. The standard way to find the GCD of two numbers is using the Euclidean algorithm, right? So maybe I can apply that here by finding the GCD of two numbers first and then using that result with the third number.

Let me start by recalling the Euclidean algorithm. It involves dividing the larger number by the smaller one, then replacing the larger number with the smaller number and the smaller number with the remainder from the division. You repeat this process until the remainder is zero, and the last non-zero remainder is the GCD. So, I need to apply this method to each pair of numbers.

First, maybe I should find GCD(3339, 2961), and then find the GCD of that result with 1491.

Let me try that.

Starting with 3339 and 2961.

<prompt_1.1> Use the Euclidean algorithm to find the GCD of 3339 and 2961 as per the above plan. </prompt_1.1>
-----

Alternatively, maybe I should check another order. Let me try GCD(2961,1491) first, then GCD that with 3339. Maybe that will confirm.

<prompt_1.2> Use the Euclidean algorithm to find the GCD of 2961 and 1491, then find the GCD with 3339. </prompt_1.2>
-----

Alternatively, maybe I can check using prime factorization. Let me try that as a cross-verification.

Starting with 1491. Let's factorize 1491.

<prompt_1.3> Use prime factorization to verify the GCD of 3339, 2961, and 1491. </prompt_1.3>
</Plan_1>

<Execution_1>
<execution_1.1>
So, divide 3339 by 2961. Let me compute that.

3339 ÷ 2961. How many times does 2961 go into 3339? Once, because 2961*1 = 2961. Subtract that from 3339: 3339 - 2961 = 378. So the remainder is 378.

Now, take 2961 and divide by 378. Let's see how many times 378 fits into 2961. Let me compute 378*7 = 2646. 2961 - 2646 = 315. So that's 7 times with a remainder of 315.

Next, divide 378 by 315. 315*1 = 315. Subtract that from 378: 378 - 315 = 63. Remainder is 63.

Now divide 315 by 63. 63*5 = 315. Exactly, no remainder. So the last non-zero remainder is 63. Therefore, GCD(3339, 2961) is 63.
</execution_1.1>

<execution_1.2>
So, GCD(2961, 1491):

2961 divided by 1491 is 1 with remainder 1470 (since 1491*1 = 1491; 2961 - 1491 = 1470).

Then GCD(1491, 1470).

1491 divided by 1470 is 1 with remainder 21 (1491 - 1470 = 21).

GCD(1470, 21).

1470 divided by 21 is 70 with remainder 0. So GCD is 21.

Now, take that 21 and find GCD(21, 3339).

3339 divided by 21. Let's calculate: 21*150 = 3150. 3339 - 3150 = 189. 189 divided by 21 is 9. So 21*159 = 3339. So 21 divides 3339 exactly. Therefore, GCD(21, 3339) is 21.

So, regardless of the order, the GCD is 21.
</execution_1.2>

<execution_1.3>
1491 ÷ 3 = 497. So 3 is a factor. Then 497: let's check divisibility. 497 ÷ 7 = 71, since 7*70=490, 490+7=497. So 497=7*71. Therefore, 1491=3*7*71.

Now 2961. Let's factorize 2961.

2961 ÷ 3 = 987. 987 ÷ 3 = 329. 329 ÷ 7 = 47. So 2961=3*3*7*47.

3339 ÷3=1113. 1113 ÷3=371. 371 ÷7=53. So 3339=3*3*7*53.

So the prime factors:

1491: 3, 7, 71

2961: 3², 7, 47

3339: 3², 7, 53

The common prime factors are 3 and 7. The lowest powers are 3^1 and 7^1. So GCD is 3*7=21.
</execution_1.3>
</Execution_1>

<Plan_2>
Based on execution_1.1:
Now I need to find GCD(63, 1491). Let's do that.

<prompt_2.1> Use the Euclidean algorithm to find the GCD of 63 and 1491. </prompt_2.1>
-----

Based on execution_1.3:
Therefore, the answer is 21. But just to make sure, let me check once again with another approach.

<prompt_2.2> Confirm the GCD of 3339, 2961, and 1491 using another approach as per the above plan. </prompt_2.2>
</Plan_2>

<Execution_2>
<execution_2.1>
So, divide 1491 by 63. Let's calculate how many times 63 goes into 1491. 63*20 = 1260. 1491 - 1260 = 231. 63*3 = 189. 231 - 189 = 42. So that's 20 + 3 = 23 times with a remainder of 42.

Now take 63 and divide by 42. 42*1 = 42. Subtract that from 63: remainder 21.

Divide 42 by 21. That's exactly 2 times with no remainder. So the GCD is 21.
</execution_2.1>

<execution_2.2>
Alternatively, perhaps using prime factorization for all three numbers:

3339: 3² * 7 * 53

2961: 3² * 7 * 47

1491: 3 * 7 * 71

So the common prime factors with minimum exponents are 3^1 and 7^1, so 3*7=21. Yep, same result.

Alternatively, if I use the method of finding GCD of all three numbers by prime factorizing each and taking the intersection primes, which is what I did. So, seems solid.

Therefore, I think the answer is 21. I don't see any mistakes in my calculations. All steps check out.
</execution_2.2>
</Execution_2>

<Plan_3>
Based on execution_2.1:
Wait, let me verify that. Let me check if 21 divides all three numbers: 3339, 2961, and 1491.

<prompt_3.1> Verify that 21 divides 3339, 2961, and 1491. </prompt_3.1>
-----

Based on execution_2.2:
**Final Answer**
The greatest common divisor is \boxed{21}.
</Plan_3>

<Execution_3>
<execution_3.1>
3339 ÷ 21: 21*150 = 3150. 3339 - 3150 = 189. 189 ÷ 21 = 9. So 150 + 9 = 159. 21*159 = 3339. Correct.

2961 ÷ 21: 21*140 = 2940. 2961 - 2940 = 21. So 140 + 1 = 141. 21*141 = 2961. Correct.

1491 ÷ 21: 21*70 = 1470. 1491 - 1470 = 21. 70 + 1 = 71. 21*71 = 1491. Correct.

So, yes, 21 is the GCD.
</execution_3.1>
</Execution_3>

<Final_answer>
Therefore, the GCD of $3339$, $2961$, and $1491$ is $\boxed{21}$.
</Final_answer>
</think>