<Question>
Express $5^5\div5^4-5^3+5^2\cdot5^1$ as an integer.
</Question>

<think>
<Plan_1>
Okay, let's see. I need to calculate this expression: 5^5 ÷ 5^4 - 5^3 + 5^2 · 5^1. Hmm, right. And express it as an integer. Alright, let's break it down step by step. Maybe I should handle each term separately first, then combine them. Let me recall the order of operations—PEMDAS, right? Parentheses, Exponents, Multiplication and Division (from left to right), Addition and Subtraction (from left to right).

So, looking at the expression: 5^5 ÷ 5^4 - 5^3 + 5^2 · 5^1. Let's identify the operations. The first term is division: 5^5 divided by 5^4. Then subtraction of 5^3, then addition of the product of 5^2 and 5^1. Let me compute each part one by one.

Starting with 5^5 ÷ 5^4. I remember that when you divide exponents with the same base, you subtract the exponents. So 5^(5-4) = 5^1 = 5. That seems right. Alternatively, I could compute each power separately: 5^5 is 3125, and 5^4 is 625. Then 3125 ÷ 625 is 5. Yep, same result.

Next term is -5^3. Wait, is that negative 5 cubed, or is the exponent applying to something else?

<prompt_1.1> Simplify the term -5^3 using the correct order of operations. </prompt_1.1>

Next term is 5^2 · 5^1. Again, same base. When multiplying exponents with the same base, add the exponents.
So 5^(2+1) = 5^3 = 125. Let me check: 5^2 is 25, 5^1 is 5, 25*5=125. Correct.

So putting it all together: the first term is 5, the second term is -125, the third term is +125. So the expression becomes 5 - 125 + 125.

Wait, let me make sure I didn't miss any signs. The original expression is 5^5 ÷ 5^4 - 5^3 + 5^2 · 5^1.
So after simplifying each term, it's 5 - 125 + 125. Now, performing the operations from left to right. So first 5 - 125 = -120. Then -120 + 125 = 5.

Alternatively, maybe someone could group the last two terms differently, but according to order of operations, since addition and subtraction are at the same level, we do them left to right.
So yes, 5 - 125 is -120, then plus 125 gives 5.

Wait, but let me double-check all steps to be sure. Maybe I made a mistake in simplifying the exponents.

<prompt_1.2> Re-evaluate each term of the expression to ensure no mistakes were made in simplification. </prompt_1.2>
-----

Alternatively, maybe there's another way to look at the expression. Let me try simplifying the entire expression using exponent rules without calculating each term separately.

<prompt_1.3> Explore an alternative method to simplify the expression using exponent rules. </prompt_1.3>
</Plan_1>

<Execution_1>
<execution_1.1>
Just the third term is addition, the first two are subtraction and addition. So 5^3 is 5*5*5=125, so this term is -125.
</execution_1.1>

<execution_1.2>
First term: 5^5 ÷ 5^4. As I said, 5^(5-4) = 5^1 = 5. Correct. 

Second term: -5^3. The exponent here is 3, so 5 cubed is 125, so this term is -125. Correct. 

Third term: 5^2 · 5^1. As exponents, 2 and 1. Adding them gives 3, so 5^3 = 125. Correct.
</execution_1.2>

<execution_1.3>
Original expression: 5^5 ÷ 5^4 - 5^3 + 5^2 · 5^1.

Simplify each term using exponent laws:

First term: 5^(5-4) = 5^1 =5.

Second term: -5^3 = -125.

Third term: 5^(2+1) =5^3=125.

So same as before: 5 -125 +125=5.
</execution_1.3>
</Execution_1>

<Plan_2>
Based on execution_1.1:
Then the third term is +5^2 ·5^1, which is +125.
So 5 -125 +125. Yep, that's 5 plus (-125 +125). But -125 +125 is 0, so 5 +0=5. Wait, so is the answer 5?

But let me verify again. Maybe I missed a sign or miscalculated.

<prompt_2.1> Re-evaluate the expression 5 - 125 + 125 to ensure the result is correct. </prompt_2.1>
-----

Alternatively, if someone does 5 - (125 +125), that would be wrong, but according to order of operations, addition and subtraction are left associative, so it's ((5 -125) +125). Which is indeed 5.
</Plan_2>

<Execution_2>
<execution_2.1>
Let me recompute each term:

5^5 is 3125, divided by 5^4 which is 625. 3125 ÷625: 625*5=3125, so yes, 5. Then subtract 5^3 which is 125. Then add 5^2*5^1=25*5=125. So 5 -125 +125.
</execution_2.1>
</Execution_2>

<Plan_3>
Based on execution_2.1:
Wait, but maybe there's a mistake in the order. Let me check the original expression again.

<prompt_3.1> Re-check the original expression and the order of operations to ensure the correct approach is used. </prompt_3.1>
-----

Based on execution_1.3:
Another check: Maybe compute all terms in decimal to verify.

5^5=3125, 5^4=625.

<prompt_3.2> Perform a final verification of the solution using decimal computation and conclude the result. </prompt_3.2>
</Plan_3>

<Execution_3>
<execution_3.1>
The original expression: 5^5 ÷5^4 -5^3 +5^2 ·5^1. So division comes first, then subtraction, then addition. So yes, 5 -125 +125. Which is indeed 5. Hmm. That seems straightforward.

Wait, but sometimes people might make a mistake in the order, but here it's clear. Division first, then left to right for subtraction and addition.
</execution_3.1>

<execution_3.2>
3125 ÷625=5. 

5^3=125. 

5^2=25, 5^1=5, 25*5=125. 

So 5 -125 +125. 

Yes, 5 -125 is -120, then -120 +125=5. 

I think that's solid. Maybe try plugging into a calculator step by step:

First, 5^5 ÷5^4. Enter 5^5: 3125. Divide by 5^4: 625. 3125 ÷625=5. 

Then subtract 5^3: 5^3=125. 5 -125= -120. 

Then add 5^2*5^1: 25*5=125. -120 +125=5. 

Yep, same result. 

So the final answer is 5. I don't see any mistakes in the steps.
</execution_3.2>
</Execution_3>

<Final_answer>
Therefore, the expression $5^5\div 5^4 -5^3 +5^2\cdot 5^1$ simplifies to $\boxed{5}$.
</Final_answer>
</think>