Okay, let's see. The problem is about finding Rosa's age based on some division remainders and her age being less than 75. Hmm. Let me break this down step by step.

First, the problem states that when Rosa's age is divided by 2, 3, 4, or 6, the remainder is 1. That means her age is 1 more than a multiple of each of these numbers. So, if I think about it, her age minus 1 should be divisible by 2, 3, 4, and 6. Therefore, Rosa's age minus 1 is a common multiple of these numbers. The least common multiple (LCM) of 2, 3, 4, and 6 would give me the smallest such number. Let me calculate that.

The prime factors:
- 2 is prime.
- 3 is prime.
- 4 is 2².
- 6 is 2×3.

The LCM would be the product of the highest powers of all primes involved. So, 2² × 3 = 4 × 3 = 12. So the LCM is 12. That means Rosa's age is 12k + 1, where k is a positive integer. So possible ages are 13, 25, 37, 49, 61, 73, 85, etc. But wait, the problem also says she's less than 75, so the possible ages would be 13, 25, 37, 49, 61, 73. Let me list them out:

13, 25, 37, 49, 61, 73.

Now, the second condition is that when her age is divided by 7, the remainder is 0. So her age must be a multiple of 7. Let's check which of the numbers above are multiples of 7.

13 divided by 7 is 1 with remainder 6. Not a multiple.
25 divided by 7 is 3 with remainder 4. Not a multiple.
37 divided by 7 is 5 with remainder 2. Not a multiple.
49 divided by 7 is 7 with remainder 0. So 49 is a multiple of 7.
61 divided by 7 is 8 with remainder 5. Not a multiple.
73 divided by 7 is 10 with remainder 3. Not a multiple.

So the only number in the list that's a multiple of 7 is 49. Therefore, Rosa must be 49 years old. But wait, let me verify this again to make sure I didn't miss anything.

Wait, 49 divided by 2: 49/2 is 24 with remainder 1. Good. Divided by 3: 49/3 is 16 with remainder 1. Divided by 4: 49/4 is 12 with remainder 1. Divided by 6: 49/6 is 8 with remainder 1. All conditions satisfied. Divided by 7 gives 7 with remainder 0. And 49 is less than 75. So yes, that works.

But let me check if there are any other possible numbers. The next possible age would be 12*(k+1) +1 = 12*7 +1 = 85, which is over 75, so that's out. So 49 is the only one. Therefore, the answer should be 49.

Wait, just to make sure I didn't make a mistake in calculating the LCM. Let me check again. The numbers 2, 3, 4, 6. Let's list their multiples:

Multiples of 2: 2,4,6,8,10,12,14,16,18,20,22,24,26,28,30,32,34,36,38,40,42,44,46,48,50,52,54,56,58,60,62,64,66,68,70,72,74,76,...

Multiples of 3: 3,6,9,12,15,18,21,24,27,30,33,36,39,42,45,48,51,54,57,60,63,66,69,72,75,...

Multiples of 4: 4,8,12,16,20,24,28,32,36,40,44,48,52,56,60,64,68,72,76,...

Multiples of 6: 6,12,18,24,30,36,42,48,54,60,66,72,78,...

So the common multiples of 2,3,4,6 are 12,24,36,48,60,72,84,... So the LCM is indeed 12. Therefore, the ages are 12k +1. Then, checking for multiples of 7, only 49 fits. So yes, 49 is the answer. I think that's correct.
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