Okay, so I need to find the greatest common factor (GCF) of 6432 and 132, and then add 11 to it. Let me start by recalling how to find the GCF of two numbers. The most reliable method is the Euclidean algorithm, which involves dividing the larger number by the smaller one and then repeating the process with the remainder until the remainder is zero. The last non-zero remainder is the GCF. Let me try to apply that here.

First, I have the two numbers: 6432 and 132. Since 6432 is much larger than 132, I'll start by dividing 6432 by 132. Let me do that division. Hmm, 132 times 48 is 6336. Let me check: 132*40 is 5280, 132*8 is 1056, so 5280 + 1056 = 6336. Then 6432 - 6336 = 96. So the remainder is 96. That means GCF(6432, 132) is the same as GCF(132, 96). 

Now, I need to find GCF(132, 96). Let's apply the Euclidean algorithm again. Divide 132 by 96. 96*1 = 96, subtract that from 132, which leaves 36. So the remainder is 36. Now, GCF(132, 96) = GCF(96, 36). 

Next step: GCF(96, 36). Divide 96 by 36. 36*2 = 72. Subtract that from 96: 96 - 72 = 24. Remainder is 24. So now, GCF(96, 36) becomes GCF(36, 24). 

Continuing: GCF(36, 24). Divide 36 by 24. 24*1 = 24. Subtract: 36 - 24 = 12. Remainder is 12. So GCF(36, 24) = GCF(24, 12). 

Then, GCF(24, 12). Dividing 24 by 12 gives a remainder of 0. So the last non-zero remainder is 12. Therefore, the GCF of 6432 and 132 is 12. 

Wait, let me verify that to make sure I didn't make a mistake in the steps. Let's check each division:

First, 6432 ÷ 132. As I calculated earlier, 132*48 = 6336, remainder 96. Correct. Then 132 ÷ 96. 96*1 = 96, remainder 36. Correct. Then 96 ÷ 36. 36*2 = 72, remainder 24. Correct. Then 36 ÷ 24. 24*1 = 24, remainder 12. Correct. Then 24 ÷ 12. Remainder 0. So yes, GCF is 12. 

Alternatively, maybe I can factor both numbers into their prime factors and multiply the common ones. Let's try that method as a cross-check. 

Starting with 132. Prime factors of 132: 132 ÷ 2 = 66, 66 ÷ 2 = 33, 33 ÷ 3 = 11. So prime factors are 2^2 * 3^1 * 11^1. 

Now 6432. Let's factor that. 6432 ÷ 2 = 3216, ÷2 = 1608, ÷2 = 804, ÷2 = 402, ÷2 = 201. So that's 2^5 so far. Then 201 ÷ 3 = 67. 67 is a prime number. So 6432 factors into 2^5 * 3^1 * 67^1. 

Comparing the prime factors of 132 (2^2, 3^1, 11^1) and 6432 (2^5, 3^1, 67^1). The common prime factors are 2 and 3. The minimum exponents for 2 are 2, and for 3 is 1. So GCF is 2^2 * 3^1 = 4*3 = 12. Yep, same result. 

So the GCF is indeed 12. Then the question says to increase this GCF by 11. So 12 + 11 = 23. Therefore, the result is 23. 

Wait, let me make sure I didn't misread the question. It says "the greatest common factor of 6432 and 132 is increased by 11". So GCF is 12, add 11 gives 23. That seems straightforward. 

But just to be thorough, maybe check another method. For example, list out all the factors of both numbers and find the greatest common one. But that would be time-consuming, but perhaps for a smaller number. Let's see.

Factors of 132: 1, 2, 3, 4, 6, 11, 12, 22, 33, 44, 66, 132.

Factors of 6432: Well, since 6432 is a multiple of 132 (as 132*48=6336, and 6432-6336=96, so not exactly a multiple, but 6432 ÷ 132 is 48.727... Wait, that's not an integer. Wait, but earlier we found GCF is 12, so 132 is 12*11, 6432 is 12*536. So the factors of 6432 would include all the factors of 12 multiplied by the factors of 536. But perhaps listing all factors is not practical here.

But since we've already confirmed via two methods (Euclidean algorithm and prime factorization) that GCF is 12, and the answer is 12 + 11 = 23, I think that's solid. 

Wait, just to check once more if GCF is indeed 12. Let's divide both numbers by 12 and see if they are integers. 

6432 ÷ 12: 12*500 = 6000, 6432 - 6000 = 432. 432 ÷ 12 = 36. So total is 500 + 36 = 536. So 6432 ÷ 12 = 536. 

132 ÷ 12 = 11. So yes, both are integers. So 12 is a common factor. Is there a larger common factor? Let's see. The next possible common factor would be 24. Let's check 6432 ÷ 24. 24*200 = 4800, 6432 - 4800 = 1632. 24*68 = 1632. So 200 + 68 = 268. So 24*268 = 6432. Then 132 ÷ 24 = 5.5, which is not an integer. So 24 is not a common factor. Similarly, 132 ÷ 6 = 22, which is integer, but 6432 ÷6=1072, which is integer, but 6 is less than 12, so 12 is the GCF. 

Therefore, the answer is 12 + 11 = 23.
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