<Question>
If Rosa's age is divided by 2, 3, 4, or 6, the remainder is 1. If her age is divided by 7, the remainder is 0. She is less than 75 years old. How many years old is Rosa?
</Question>

<think>
<Plan_1>
Okay, let me try to figure out Rosa's age. The problem says that when her age is divided by 2, 3, 4, or 6, the remainder is 1 each time. But when she divides her age by 7, there's no remainder. Also, she's less than 75 years old. Hmm, so I need to find a number that meets all these conditions.

First, let's break down the problem. The key here is that her age leaves a remainder of 1 when divided by 2, 3, 4, and 6. That means her age is 1 more than a multiple of each of these numbers. So if I subtract 1 from her age, the result should be divisible by 2, 3, 4, and 6 without any remainder. Then, her age must be a multiple of 7. So, essentially, I need to find a number less than 75 that is 1 more than a common multiple of 2, 3, 4, and 6, and also a multiple of 7.

Let me start by finding the least common multiple (LCM) of 2, 3, 4, and 6. The LCM of these numbers will give me the smallest number that is divisible by all of them. Then, adding 1 to the LCM will give me the smallest possible age that satisfies the remainder conditions. After that, I can check multiples of that LCM plus 1 to see which one is also a multiple of 7 and under 75.

Calculating LCM of 2, 3, 4, 6. Let's factor each number:

<prompt_1.1> Calculate the LCM of 2, 3, 4, and 6 by factoring each number and finding the product of the highest powers of all primes present. </prompt_1.1>
</Plan_1>

<Execution_1>
<execution_1.1>
- 2 is prime: 2
- 3 is prime: 3
- 4 is 2²
- 6 is 2 × 3

The LCM is the product of the highest powers of all primes present. So that would be 2² (from 4), 3 (from 3 or 6). Therefore, LCM = 2² × 3 = 4 × 3 = 12.
</execution_1.1>
</Execution_1>

<Plan_2>
Based on execution_1.1:
So, the LCM is 12. That means any number that is a multiple of 12 will be divisible by 2, 3, 4, and 6. Therefore, if we add 1 to this LCM, we get 13. So, 13 is a number that when divided by 2, 3, 4, or 6 gives a remainder of 1. Let's check:

<prompt_2.1> Verify that the number 13 satisfies the remainder conditions when divided by 2, 3, 4, and 6. </prompt_2.1>

But Rosa is less than 75, so there could be larger numbers. The next number would be 12 × 2 + 1 = 25. Let's check 25:

<prompt_2.2> Verify that the number 25 satisfies the remainder conditions when divided by 2, 3, 4, and 6, and check divisibility by 7. </prompt_2.2>

Then 37:

<prompt_2.3> Verify that the number 37 satisfies the remainder conditions when divided by 2, 3, 4, and 6, and check divisibility by 7. </prompt_2.3>

Next, 49:

<prompt_2.4> Verify that the number 49 satisfies the remainder conditions when divided by 2, 3, 4, and 6, and check divisibility by 7. </prompt_2.4>

Next would be 61:

<prompt_2.5> Verify that the number 61 satisfies the remainder conditions when divided by 2, 3, 4, and 6, and check divisibility by 7. </prompt_2.5>

Then 73:

<prompt_2.6> Verify that the number 73 satisfies the remainder conditions when divided by 2, 3, 4, and 6, and check divisibility by 7. </prompt_2.6>
</Plan_2>

<Execution_2>
<execution_2.1>
13 ÷ 2 = 6 with remainder 1 ✔️
13 ÷ 3 = 4 with remainder 1 ✔️
13 ÷ 4 = 3 with remainder 1 ✔️
13 ÷ 6 = 2 with remainder 1 ✔️
</execution_2.1>

<execution_2.2>
- 25 ÷ 2 = 12 remainder 1 ✔️
- 25 ÷ 3 = 8 remainder 1 ✔️
- 25 ÷ 4 = 6 remainder 1 ✔️
- 25 ÷ 6 = 4 remainder 1 ✔️

But 25 divided by 7 is 3 with a remainder of 4. So that's not divisible by 7.
</execution_2.2>

<execution_2.3>
25 ÷ 2 = 12 remainder 1 ✔️
25 ÷ 3 = 8 remainder 1 ✔️
25 ÷ 4 = 6 remainder 1 ✔️
25 ÷ 6 = 4 remainder 1 ✔️

And 25 ÷ 7 = 3 with remainder 4. So that's not divisible by 7. Okay, so 25 is out.
</execution_2.3>

<execution_2.4>
25 ÷ 2 = 12 remainder 1

25 ÷ 3 = 8 remainder 1

25 ÷ 4 = 6 remainder 1

25 ÷ 6 = 4 remainder 1. Okay, so 25 works. Then 37:

37 ÷ 2 = 18 remainder 1

37 ÷ 3 = 12 remainder 1

37 ÷ 4 = 9 remainder 1

37 ÷ 6 = 6 remainder 1. Good.

49 ÷ 2 = 24 remainder 1

49 ÷ 3 = 16 remainder 1

49 ÷ 4 = 12 remainder 1

49 ÷ 6 = 8 remainder 1. Also works.

61 ÷ 2 = 30 remainder 1

61 ÷ 3 = 20 remainder 1

61 ÷ 4 = 15 remainder 1

61 ÷ 6 = 10 remainder 1. Yep.

73 ÷ 2 = 36 remainder 1

73 ÷ 3 = 24 remainder 1

73 ÷ 4 = 18 remainder 1

73 ÷ 6 = 12 remainder 1. That works too.

So the possible numbers under 75 are 13, 25, 37, 49, 61, 73.
</execution_2.4>

<execution_2.5>
25 ÷ 2 = 12 remainder 1 ✔️

25 ÷ 3 = 8 remainder 1 ✔️

25 ÷ 4 = 6 remainder 1 ✔️

25 ÷ 6 = 4 remainder 1 ✔️

Okay, so 25 works. Then 37:

37 ÷ 2 = 18 r1 ✔️

37 ÷ 3 = 12 r1 ✔️

37 ÷ 4 = 9 r1 ✔️

37 ÷ 6 = 6 r1 ✔️

Good. 37 is next. Then 49:

49 ÷ 2 = 24 r1 ✔️

49 ÷ 3 = 16 r1 ✔️

49 ÷ 4 = 12 r1 ✔️

49 ÷ 6 = 8 r1 ✔️

Okay, 49 is next. Then 61:

61 ÷ 2 = 30 r1 ✔️

61 ÷ 3 = 20 r1 ✔️

61 ÷ 4 = 15 r1 ✔️

61 ÷ 6 = 10 r1 ✔️

61 is under 75. Then 73:

73 ÷ 2 = 36 r1 ✔️

73 ÷ 3 = 24 r1 ✔️

73 ÷ 4 = 18 r1 ✔️

73 ÷ 6 = 12 r1 ✔️

73 is also under 75. The next one would be 85, which is over 75, so we can stop here.
</execution_2.5>

<execution_2.6>
Check: 61 ÷ 2 = 30 remainder 1, 61 ÷ 3 = 20 remainder 1, 61 ÷ 4 = 15 remainder 1, 61 ÷ 6 = 10 remainder 1. Then 73 ÷ 2 = 36 rem 1, 73 ÷ 3 = 24 rem 1, 73 ÷ 4 = 18 rem 1, 73 ÷ 6 = 12 rem 1. All good. But 73 is less than 75. Wait, but 73 is the next one after 61. Then 73. Then 73 +12=85, but 85 is over 75, so stop here.
</execution_2.6>
</Execution_2>

<Plan_3>
Based on execution_2.1, execution_2.2, execution_2.3, execution_2.4, execution_2.5, execution_2.6:
Now, from these numbers: 13, 25, 37, 49, 61, 73, we need to find which one is divisible by 7.

Let's check each one:

<prompt_3.1> Check which of the numbers 13, 25, 37, 49, 61, 73 is divisible by 7 and verify the solution. </prompt_3.1>
</Plan_3>

<Execution_3>
<execution_3.1>
13 ÷ 7 = 1 with remainder 6. Not divisible by 7.

25 ÷ 7 = 3 with remainder 4. Not divisible by 7.

37 ÷ 7 = 5 with remainder 2. Not divisible by 7.

49 ÷ 7 = 7 exactly. No remainder. So 49 is divisible by 7. That seems to fit. Then 61 ÷ 7 = 8 with remainder 5. Not divisible by 7. 73 ÷ 7 = 10 with remainder 3. Not divisible by 7. So only 49 is divisible by 7.

Wait, but let me check again. Maybe I made a mistake in checking 49.
</execution_3.1>
</Execution_3>

<Plan_4>
Based on execution_3.1:
49 ÷ 7 is 7, so yes, that's correct. So 49 is a candidate.

But wait, let me verify all the conditions for 49.

<prompt_4.1> Verify that Rosa's age is 49 by checking all the given conditions. </prompt_4.1>
</Plan_4>

<Execution_4>
<execution_4.1>
First, divided by 2: 49 ÷ 2 = 24 remainder 1. Correct.

Divided by 3: 49 ÷ 3 = 16 remainder 1. Correct.

Divided by 4: 49 ÷ 4 = 12 remainder 1. Correct.

Divided by 6: 49 ÷ 6 = 8 remainder 1. Correct.

Divided by 7: 49 ÷ 7 = 7 exactly. No remainder. Correct.

And 49 is less than 75. Yes. So 49 meets all the criteria.

But wait, let me check if there are any other numbers in the list that also satisfy the divisibility by 7. The next one after 49 would be 49 + LCM(12,7). Since LCM(12,7) is 84, because 12 and 7 are coprime. So 84 is the LCM. So next number would be 49 + 84 = 133, which is over 75, so there are no other numbers in the range. Therefore, 49 is the only one.
</execution_4.1>
</Execution_4>

<Plan_5>
Based on execution_4.1:
But wait, let me double-check. Maybe I missed something. Let me verify 49 again.

<prompt_5.1> Double-check the solution by re-evaluating the conditions for Rosa's age. </prompt_5.1>
</Plan_5>

<Execution_5>
<execution_5.1>
49 divided by 2 is 24.5, but since we're dealing with whole numbers, 24*2=48, 49-48=1. So remainder 1. Same for 3: 3*16=48, 49-48=1. For 4: 4*12=48, remainder 1. For 6: 6*8=48, remainder 1. All correct. Divided by 7: 7*7=49, no remainder. And 49 is less than 75. So yes, 49 seems to fit all the conditions.
</execution_5.1>
</Execution_5>

<Plan_6>
Based on execution_5.1:
But wait, let me check if there's another possible age. For example, maybe I missed a multiple.

<prompt_6.1> Confirm that there is only one solution for Rosa's age by checking all possibilities. </prompt_6.1>
</Plan_6>

<Execution_6>
<execution_6.1>
Wait, the numbers we had were 13,25,37,49,61,73. Only 49 is divisible by 7. So 49 is the only one. Therefore, the answer is 49.

But let me think again. Maybe there's another way to approach this problem. Let's see. Since her age is divisible by 7, let's list all multiples of 7 under 75 and check which ones leave a remainder of 1 when divided by 2, 3, 4, and 6.

So multiples of 7 under 75: 7, 14, 21, 28, 35, 42, 49, 56, 63, 70, 77 (but 77 is over 75, so stop at 70).

Now check each of these to see if they leave remainder 1 when divided by 2, 3, 4, 6.

Starting with 7:

7 ÷ 2 = 3 rem 1 ✔️

7 ÷ 3 = 2 rem 1 ✔️

7 ÷ 4 = 1 rem 3 ❌

So nope, doesn't work.

14:

14 ÷ 2 = 7 rem 0 ❌

So already fails.

21:

21 ÷ 2 = 10 rem 1 ✔️

21 ÷ 3 = 7 rem 0 ❌

Fails.

28:

28 ÷ 2 = 14 rem 0 ❌

Nope.

35:

35 ÷ 2 = 17 rem 1 ✔️

35 ÷ 3 = 11 rem 2 ❌

Fails.

42:

42 ÷ 2 = 21 rem 0 ❌

Nope.

49:

49 ÷ 2 = 24 rem 1 ✔️

49 ÷ 3 = 16 rem 1 ✔️

49 ÷ 4 = 12 rem 1 ✔️

49 ÷ 6 = 8 rem 1 ✔️

All conditions met. So 49 is good.

56:

56 ÷ 2 = 28 rem 0 ❌

Nope.

63:

63 ÷ 2 = 31 rem 1 ✔️

63 ÷ 3 = 21 rem 0 ❌

Fails.

70:

70 ÷ 2 = 35 rem 0 ❌

Nope.

So again, only 49 fits all the criteria. Therefore, the answer must be 49.
</execution_6.1>
</Execution_6>

<Plan_7>
Based on execution_6.1:
Just to ensure there's no mistake, let's verify once more:

<prompt_7.1> Perform a final verification of the solution and conclude Rosa's age. </prompt_7.1>
</Plan_7>

<Execution_7>
<execution_7.1>
49 divided by 2: 2*24=48, 49-48=1. Correct.

Divided by 3: 3*16=48, 49-48=1. Correct.

Divided by 4: 4*12=48, 49-48=1. Correct.

Divided by 6: 6*8=48, 49-48=1. Correct.

Divided by 7: 7*7=49. No remainder. Correct.

Age <75: 49 is less than 75. Correct.

Therefore, all conditions are satisfied. I don't see any mistakes in the reasoning. So Rosa must be 49 years old.
</execution_7.1>
</Execution_7>

<Final_answer>
Therefore, Rosa is $\boxed{49}$ years old.
</Final_answer>
</think>