Okay, so I need to find the greatest common divisor (GCD) of three numbers: 3339, 2961, and 1491. Hmm, GCD is the largest number that divides all three numbers without leaving a remainder. Let me think about the best way to approach this.

First, I remember that the GCD of multiple numbers can be found by finding the GCD of pairs step by step. So maybe I can start by finding the GCD of two numbers first and then use that result to find the GCD with the third number. That seems manageable.

Let me start with the first two numbers: 3339 and 2961. To find their GCD, the Euclidean algorithm is the standard method. Let me recall how that works. The Euclidean algorithm 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, repeating this process until the remainder is zero. The last non-zero remainder is the GCD.

So, let's apply that to 3339 and 2961.

First, divide 3339 by 2961. Let me calculate how many times 2961 fits into 3339.

2961 * 1 = 2961

Subtract that from 3339: 3339 - 2961 = 378. So the remainder is 378.

Now, replace the numbers: GCD(2961, 378).

Next, divide 2961 by 378. Let's see how many times 378 goes into 2961.

378 * 7 = 2646

Subtract that from 2961: 2961 - 2646 = 315. Remainder is 315.

Now, GCD(378, 315).

Divide 378 by 315. 315 * 1 = 315. Subtract from 378: 378 - 315 = 63. Remainder is 63.

Now, GCD(315, 63).

Divide 315 by 63. 63 * 5 = 315. Remainder is 0.

So the GCD of 3339 and 2961 is 63.

Now, I need to find the GCD of this result, 63, with the third number, 1491.

Again, using the Euclidean algorithm. So GCD(63, 1491).

Divide 1491 by 63. Let's calculate how many times 63 fits into 1491.

63 * 20 = 1260

1491 - 1260 = 231. So remainder is 231.

Now, GCD(63, 231).

Divide 231 by 63. 63 * 3 = 189. Subtract from 231: 231 - 189 = 42. Remainder is 42.

Now, GCD(63, 42).

Divide 63 by 42. 42 * 1 = 42. Subtract from 63: 63 - 42 = 21. Remainder is 21.

Now, GCD(42, 21).

Divide 42 by 21. 21 * 2 = 42. Remainder is 0.

So the GCD of 63 and 1491 is 21.

Therefore, the GCD of all three numbers 3339, 2961, and 1491 is 21.

Wait, let me verify this to make sure I didn't make a mistake. Let me check each step again.

First, GCD(3339, 2961):

3339 ÷ 2961 = 1 with remainder 378. Correct.

2961 ÷ 378 = 7 with remainder 315. 378*7=2646, 2961-2646=315. Correct.

378 ÷ 315 = 1 with remainder 63. Correct.

315 ÷ 63 = 5 with remainder 0. So GCD is 63. That's right.

Then GCD(63, 1491):

1491 ÷ 63 = 23 with remainder 231. Because 63*23=1449, 1491-1449=42. Wait, wait, hold on. Wait, 63*23: 60*23=1380, 3*23=69, so total 1380+69=1449. Then 1491-1449=42. So remainder is 42, not 231. Wait, did I make a mistake here?

Wait, I think I messed up the division step. Let me recalculate 1491 ÷ 63.

63*20=1260. 1491-1260=231. So 231 ÷63. 63*3=189. 231-189=42. So remainder is 42. So GCD(63,231) becomes GCD(63,42). Then GCD(63,42):

63 ÷42=1 with remainder 21. Then GCD(42,21)=21. So yes, the GCD is 21. So that part is correct.

But wait, when I did 1491 ÷63, I thought 63*23=1449, which is correct. 1491-1449=42. So remainder is 42. So that step is correct. Then GCD(63,42) gives 21. So the final answer is 21. So that's correct.

But let me check another way. Maybe factor each number into primes and see the common factors.

Factor 3339:

3339 ÷3=1113 (since 3+3+3+9=18, divisible by 3). 1113 ÷3=371. 371 ÷7=53. So 3339=3*3*7*53.

Factor 2961:

2961 ÷3=987 (sum of digits 2+9+6+1=18, divisible by 3). 987 ÷3=329. 329 ÷7=47. So 2961=3*3*7*47.

Factor 1491:

1491 ÷3=497. 497 ÷7=71. So 1491=3*7*71.

Now, the prime factors:

3339: 3² *7 *53

2961: 3² *7 *47

1491: 3 *7 *71

The common prime factors are 3 and 7. The minimum exponents for 3 is 1 and for 7 is 1. So GCD is 3^1 *7^1=21. So that confirms the answer.

Therefore, the greatest common divisor is indeed 21.
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