<Question>
The Fibonacci sequence is the sequence 1, 1, 2, 3, 5, $\ldots$ where each term is the sum of the previous two terms. What is the remainder when the $100^{\mathrm{th}}$ term of the sequence is divided by 4?
</Question>

<think>
<Plan_1>
Okay, so I need to find the remainder when the 100th term of the Fibonacci sequence is divided by 4. Hmm, Fibonacci sequence is that 1, 1, 2, 3, 5, 8, 13, 21, and so on, where each term is the sum of the two before it. Let me think about how to approach this.

First, I remember that when dealing with remainders, modular arithmetic can be really helpful. Maybe if I compute the Fibonacci sequence modulo 4, I can find a pattern or cycle. If there's a repeating cycle, then I can determine where the 100th term falls within that cycle and find the remainder accordingly.

Let me start by writing out the Fibonacci sequence and taking each term modulo 4. Let's see:

<prompt_1.1> Write out the Fibonacci sequence and compute each term modulo 4 to identify any repeating pattern. </prompt_1.1>
</Plan_1>

<Execution_1>
<execution_1.1>
Term 1: 1 (mod 4 is 1)
Term 2: 1 (mod 4 is 1)
Term 3: 1 + 1 = 2 (mod 4 is 2)
Term 4: 1 + 2 = 3 (mod 4 is 3)
Term 5: 2 + 3 = 5 (mod 4 is 1)
Term 6: 3 + 5 = 8 (mod 4 is 0)
Term 7: 5 + 8 = 13 (mod 4 is 1)
Term 8: 8 + 13 = 21 (mod 4 is 1)
Term 9: 13 + 21 = 34 (mod 4 is 2)
Term 10: 21 + 34 = 55 (mod 4 is 3)
Term 11: 34 + 55 = 89 (mod 4 is 1)
Term 12: 55 + 89 = 144 (mod 4 is 0)
</execution_1.1>
</Execution_1>

<Plan_2>
Based on execution_1.1:
Wait a second, let me list these out:

1: 1

<prompt_2.1> List the computed modulo 4 values and identify the repeating cycle. </prompt_2.1>
</Plan_2>

<Execution_2>
<execution_2.1>
2: 1

3: 2

4: 3

5: 1

6: 0

7: 1

8: 1

9: 2

10: 3

11: 1

12: 0

So starting from term 1, the sequence modulo 4 is: 1, 1, 2, 3, 1, 0, 1, 1, 2, 3, 1, 0...
</execution_2.1>
</Execution_2>

<Plan_3>
Based on execution_2.1:
It looks like after the first few terms, the pattern 1, 1, 2, 3, 1, 0 starts repeating. Let me check if that's consistent.

<prompt_3.1> Verify the repeating cycle by checking subsequent terms. </prompt_3.1>

So the Pisano period for modulo 4 is 6. That means every 6 terms, the sequence repeats when taken modulo 4.
Therefore, to find the 100th term modulo 4, we can find where 100 falls in the cycle. Let's calculate 100 divided by 6 and find the remainder.

<prompt_3.2> Calculate the remainder of the 100th term using the Pisano period. </prompt_3.2>
</Plan_3>

<Execution_3>
<execution_3.1>
Term 7: 1 (same as term 1)

Term 8: 1 (same as term 2)

Term 9: 2 (same as term 3)

Term 10: 3 (same as term 4)

Term 11: 1 (same as term 5)

Term 12: 0 (same as term 6)

Yep, so the cycle length here is 6. So every 6 terms, the pattern repeats.
</execution_3.1>

<execution_3.2>
6*16 = 96, so 100 - 96 = 4. So the remainder is 4. That means the 100th term corresponds to the 4th term in the cycle.

Looking back at the cycle: 1 (term 1), 1 (term 2), 2 (term 3), 3 (term 4), 1 (term 5), 0 (term 6). So the 4th term in the cycle is 3. Therefore, the remainder when the 100th Fibonacci term is divided by 4 is 3.
</execution_3.2>
</Execution_3>

<Plan_4>
Based on execution_3.2:
Wait, let me verify this.

<prompt_4.1> Verify the calculated remainder by checking previous terms in the cycle. </prompt_4.1>

But let me cross-verify by calculating a few more terms to ensure the cycle is indeed 6.

<prompt_4.2> Cross-verify the cycle and the remainder by calculating additional terms. </prompt_4.2>

But just to be thorough, let me compute the first, say, 15 terms modulo 4 to confirm the cycle.

<prompt_4.3> Compute additional terms to confirm the cycle and determine the remainder of the 100th term. </prompt_4.3>
</Plan_4>

<Execution_4>
<execution_4.1>
Let's check term 10: 55. 55 divided by 4 is 13*4=52, remainder 3. Correct. Term 16 would be 100 - 96 = 4th term in the cycle, which is 3. Let's check term 16.
</execution_4.1>

<execution_4.2>
Let's see term 13: 89 + 144 = 233. 233 mod 4: 233 divided by 4 is 58*4=232, remainder 1. So term 13 mod 4 is 1 (which is term 7 and term 1). Term 14: 144 + 233 = 377. 377 mod 4: 376 is 94*4, so remainder 1. So term 14 mod 4 is 1 (same as term 8 and term 2). So yes, the cycle does repeat every 6 terms. So the cycle length is 6.

Therefore, the Fibonacci sequence modulo 4 has a period of 6. So positions 1, 7, 13, ... all have the same modulo 4 value. Similarly for positions 2,8,14,... etc. So term 100: let's compute 100 divided by 6. 6*16=96, remainder 4. So term 100 corresponds to the 4th term in the cycle. The 4th term in the cycle is 3. So remainder 3. So the answer should be 3.
</execution_4.2>

<execution_4.3>
Let's see:

1: 1

2: 1

3: 2

4: 3

5: 1 (3+1=4 mod4=0? Wait, wait, hold on. Wait, term 5 is term 3 + term 4. So term 3 is 2, term 4 is 3. 2 + 3 = 5, which is 1 mod4. Correct.

Term 6: term4 + term5 = 3 + 1 = 4 mod4=0

Term7: term5 + term6 = 1 + 0 =1

Term8: term6 + term7=0 +1=1

Term9: term7 + term8=1+1=2

Term10: term8 + term9=1+2=3

Term11: term9 + term10=2+3=5 mod4=1

Term12: term10 + term11=3 +1=4 mod4=0

Term13: term11 + term12=1 +0=1

Term14: term12 + term13=0 +1=1

Term15: term13 + term14=1 +1=2

Term16: term14 + term15=1 +2=3

So the sequence from term1 to term16 modulo4 is:

1,1,2,3,1,0,1,1,2,3,1,0,1,1,2,3

Yep, so starting from term1, the cycle is 1,1,2,3,1,0 and repeats every 6 terms. So every 6 terms, the pattern repeats. Therefore, the Pisano period for modulo4 is indeed 6. So term1:1, term7:1, term13:1, etc. So term6k +1:1, term6k +2:1, etc.

Therefore, term 100: Let's compute 100 divided by6. 6*16=96, remainder4. So term96 +4= term100. So term96 is the last term of the 16th cycle, which is 0. Then term97:1, term98:1, term99:2, term100:3. So yes, term100 modulo4 is3. Therefore, the remainder is3.
</execution_4.3>
</Execution_4>

<Plan_5>
Based on execution_4.3:
Alternatively, maybe I can compute Fibonacci numbers up to the 100th term, but that's impractical. However, since we have the cycle of6, we can compute term1 to term6, then use the cycle.

<prompt_5.1> Confirm the solution using different methods and finalize the answer. </prompt_5.1>
</Plan_5>

<Execution_5>
<execution_5.1>
Alternatively, maybe there's a formula or a matrix exponentiation method, but for modulo4, since the cycle is short, it's easier to use the cycle.

Alternatively, another way: since the Pisano period for modulo4 is6, then F(n) mod4 = F(n mod6) mod4, but adjusted for the cycle. Wait, but the cycle starts at term1. So if n is a multiple of6, then F(n) mod4 is0. For example, term6:0, term12:0, term18:0, etc. Then term1:1, term7:1, which is F(1) mod4=1. So if n mod6=1, then1; n mod6=2, then1; n mod6=3, then2; n mod6=4, then3; n mod6=5, then1; n mod6=0, then0. So in general, F(n) mod4 can be determined by n mod6. Therefore, for n=100, compute 100 mod6.

100 divided by6 is16*6=96, remainder4. Therefore, 100 mod6=4. Then according to the mapping above, when n mod6=4, the remainder is3. Therefore, the answer is3. So this method also gives3.

Therefore, multiple methods confirm that the remainder is3. So I think that's solid. I don't see any mistakes in the reasoning. Checked the cycle manually, verified with division, and cross-validated with Pisano period concepts. So confident the answer is3.
</execution_5.1>
</Execution_5>

<Final_answer>
Therefore, the remainder when the $100^{\mathrm{th}}$ term of the sequence is divided by 4 is $\boxed{3}$.
</Final_answer>
</think>