Explanation:
Okay, so I need to figure out for which prime numbers p the congruence 8x ≡ 1 mod p has no solutions. Then, sum all those primes. Hmm, let me start by recalling some number theory concepts.

First, the congruence 8x ≡ 1 mod p is a linear congruence. I remember that a linear congruence ax ≡ b mod m has a solution if and only if gcd(a, m) divides b. In this case, a is 8, b is 1, and m is p, which is a prime. So, the condition for the congruence to have a solution is that gcd(8, p) divides 1.

But since p is a prime, let's think about what gcd(8, p) can be. The gcd of 8 and p can only be 1 or p, right? Because p is prime. If p divides 8, then gcd(8, p) is p. Otherwise, it's 1.

So, if p divides 8, then gcd(8, p) = p. But since p divides 8, p must be one of the prime factors of 8. The prime factors of 8 are just 2, since 8 is 2^3. Therefore, the only prime p for which gcd(8, p) is not 1 is p = 2.

So, for p ≠ 2, gcd(8, p) = 1, which divides 1, so the congruence 8x ≡ 1 mod p has a solution. However, for p = 2, gcd(8, 2) = 2. Then, since 2 does not divide 1 (because 1 divided by 2 is not an integer), the congruence 8x ≡ 1 mod 2 has no solution.

Wait, let me check that. Let me test p = 2. The congruence is 8x ≡ 1 mod 2. But 8 mod 2 is 0, so the equation becomes 0x ≡ 1 mod 2, which simplifies to 0 ≡ 1 mod 2. That's obviously false, so there's no solution. So p = 2 is such a prime.

But the question says "finitely many primes p" for which the congruence has no solutions. But wait, primes other than 2, the congruence has solutions? Let me check with another prime, say p = 3. Then 8x ≡ 1 mod 3. Since 8 mod 3 is 2, so 2x ≡ 1 mod 3. Then x ≡ 2 mod 3, since 2*2=4≡1 mod 3. So that works. Similarly, for p = 5: 8x ≡ 1 mod 5. 8 mod 5 is 3, so 3x ≡ 1 mod 5. Multiply both sides by inverse of 3 mod 5. The inverse of 3 mod 5 is 2 because 3*2=6≡1 mod 5. So x ≡ 2 mod 5. So that works.

So seems like only p=2 is the prime where there's no solution. But wait, the problem says "finitely many primes p", which is true because there's only one such prime. But let me check another prime, like p=7. 8x ≡1 mod 7. 8 mod 7 is 1, so 1x ≡1 mod 7, so x≡1 mod 7. So that works.

Wait, so maybe only p=2? But let me think again. The original question says "finitely many primes p", so maybe there are more? Wait, but according to the logic above, only p=2 divides 8, so only p=2. But maybe I made a mistake here?

Wait, let me think again. The congruence 8x ≡1 mod p has no solution if and only if gcd(8, p) does not divide 1. Since gcd(8, p) is either 1 or p. So, if gcd(8, p) divides 1, then since gcd(8, p) is either 1 or p, then p divides 1? Wait, no. Wait, the condition is that gcd(8, p) divides 1. So, gcd(8, p) must be a divisor of 1. The only positive integer that divides 1 is 1 itself. Therefore, gcd(8, p) must be 1. Therefore, the congruence has a solution if and only if gcd(8, p) = 1. Therefore, the congruence has no solution if and only if gcd(8, p) ≠ 1. Since p is prime, gcd(8, p) ≠ 1 only when p divides 8. Therefore, p divides 8. The primes dividing 8 are only 2. Therefore, only p=2. Therefore, the only prime is 2. Therefore, the sum of all such primes is 2.

But wait, the problem says "finitely many primes p", which is true, since there's only one. But maybe I need to check if there are other primes where 8x ≡1 mod p has no solution? Let me think again. Suppose p divides 8, so p=2. Then, as we saw, there's no solution. For other primes, since gcd(8, p)=1, so there is a solution. Therefore, only p=2. Therefore, the answer is 2.

But wait, let me check p=2 again. The equation is 8x ≡1 mod 2. Since 8 is congruent to 0 mod 2, so 0x ≡1 mod 2, which is 0≡1 mod 2, which is impossible. Therefore, no solution. So p=2 is the only prime. Therefore, the sum is 2.

But the problem says "finitely many primes p", which is true, but maybe I need to check if there are other primes? For example, suppose p divides 8? But 8 is 2^3, so only prime divisor is 2. Therefore, only p=2.

Wait, but maybe there's another way to think about this? For example, the equation 8x ≡1 mod p has a solution if and only if 8 is invertible modulo p. Which is equivalent to gcd(8, p)=1. Therefore, 8 is invertible modulo p if and only if p does not divide 8. Therefore, the equation has no solution if and only if p divides 8. Therefore, again, only p=2.

Therefore, the answer is 2. Therefore, the sum is 2.

But wait, the problem says "the sum of all such p". So if there is only one such prime, which is 2, then the sum is 2.

But maybe I made a mistake here? Let me check with p=2. Let me check if there is any x such that 8x ≡1 mod 2. Since 8x mod 2 is 0, because 8 is even. So 0 ≡1 mod 2? No. Therefore, no solution.

But maybe there's another prime? Let me think. Suppose p= something else. For example, p= 1? Wait, p is prime, so p=1 is not prime. So no.

Wait, maybe negative primes? No, primes are positive integers greater than 1.

Alternatively, maybe I need to think about the multiplicative inverse. The equation 8x ≡1 mod p has a solution if and only if 8 has a multiplicative inverse modulo p. Which is true if and only if 8 and p are coprime. Which is true if and only if p is not a divisor of 8. Since p is prime, p divides 8 only if p=2. Therefore, only p=2.

Therefore, the answer is 2. Therefore, the sum is 2.

But maybe the problem is trickier? Let me check again.

Wait, maybe there's a case when p divides 8? But p is prime. So p divides 8 implies p=2. So only p=2.

Alternatively, maybe the problem is in the wording? "the congruence 8x ≡1 mod p has no solutions x". So maybe there are other primes where 8x ≡1 mod p has no solution?

Wait, suppose p divides 8x -1. But for the equation to have no solution, there is no x such that 8x ≡1 mod p. Which is equivalent to saying that 8 is not invertible mod p. Which again is equivalent to gcd(8, p) ≠1. Which again is equivalent to p divides 8. Which is p=2.

Therefore, I think my conclusion is correct. Therefore, the only prime is 2, so the sum is 2.

But maybe the problem is expecting more primes? Let me check with p=  something else. For example, p= 17. Let me check if 8x ≡1 mod 17 has a solution. Let me compute 8^{-1} mod 17. Since 8*2=16≡-1 mod17, so 8*(-2)≡1 mod17. Therefore, -2 mod17 is 15. Therefore, x≡15 mod17. Therefore, solution exists.

Another prime, p=7. 8x≡1 mod7. 8≡1 mod7, so x≡1 mod7.

p=3: 8≡2 mod3, so 2x≡1 mod3. x≡2 mod3.

p=5: 8≡3 mod5. 3x≡1 mod5. x≡2 mod5.

p=11: 8x≡1 mod11. Let me compute inverse of 8 mod11. 8*? ≡1 mod11. 8* 7=56≡1 mod11 (since 55 is 5*11). So x≡7 mod11.

So all these primes have solutions.

So only p=2.

Therefore, the answer is 2.

But the problem says "finitely many primes p", which is true, since there's only one. Therefore, the sum is 2.

But maybe I need to check if there are other primes where 8x ≡1 mod p has no solution? For example, suppose p divides 8x -1. But for there to be no solution, there is no x such that 8x ≡1 mod p. Which would mean that 8x -1 is never divisible by p. Which would mean that 8x ≡1 mod p is impossible. Which again is equivalent to 8 not being invertible mod p. Which again is equivalent to p divides 8.

Therefore, only p=2.

Therefore, the answer is 2.

But maybe the problem is in the way the question is phrased? Let me check again.

"Determine the sum of all such p." So if there is only one such prime, which is 2, then the answer is 2.

Alternatively, maybe I made a mistake in thinking that p divides 8. Let me think again.

The condition for the congruence 8x ≡1 mod p to have no solution is that gcd(8, p) does not divide 1. Since gcd(8, p) divides 1 only if gcd(8, p)=1. Therefore, if gcd(8, p) ≠1, then the congruence has no solution. Since p is prime, gcd(8, p) is p if p divides 8, else 1. Therefore, p divides 8 implies p=2. Therefore, only p=2.

Therefore, the answer is 2.

But maybe the problem is expecting more? Wait, maybe I need to check if p divides 8x -1? But that's not the case. The congruence 8x ≡1 mod p has no solution if and only if there is no x such that 8x -1 is divisible by p. Which is equivalent to saying that 8x ≡1 mod p is impossible. Which is equivalent to 8 not being invertible mod p. Which is equivalent to gcd(8, p) ≠1. Which again is equivalent to p divides 8.

Therefore, only p=2.

Therefore, the answer is 2.

But maybe the problem is in the original question? Let me check again.

Original question: "There are finitely many primes p for which the congruence 8x≡1 mod p has no solutions x. Determine the sum of all such p."

Yes, so only p=2. Therefore, sum is 2.

But maybe I need to check if p=2 is considered? Since 2 is a prime. Yes.

Alternatively, maybe there's a mistake in my logic? Let me think again.

Suppose p is a prime. The equation 8x ≡1 mod p.

If p ≠2, then 8 and p are coprime, since p is odd. Therefore, 8 has an inverse modulo p, so there is a solution.

If p=2, then 8x ≡1 mod 2. Since 8 is even, 8x is 0 mod 2, so 0 ≡1 mod 2, which is false. Therefore, no solution.

Therefore, only p=2.

Therefore, the answer is 2.

But maybe the problem is expecting more? Let me check with p=  something else. For example, p= 8? But p must be prime. 8 is not prime.

Alternatively, maybe the problem is in the case when p divides 1? But p divides 1 only if p=1, which is not prime.

Therefore, no.

Therefore, I think my conclusion is correct. The only prime is 2, so the sum is 2.

But maybe the problem is from somewhere, and the answer is different? Wait, maybe I need to check if there are other primes where 8x ≡1 mod p has no solution.

Wait, suppose p divides 8x -1. For there to be no solution, there is no x such that p divides 8x -1. Which would mean that 8x ≡1 mod p is impossible. Which is equivalent to 8 not being invertible mod p. Which again is equivalent to gcd(8, p) ≠1. Which again is equivalent to p divides 8.

Therefore, only p=2.

Therefore, the answer is 2.

Therefore, I think the answer is 2.

**Final Answer**
The sum of all such primes is \boxed{2}.

To determine the sum of all prime numbers $ p $ for which the congruence

$$
8x \equiv 1 \pmod{p}
$$

has **no solutions**, we begin by analyzing the conditions under which such a linear congruence has a solution.

---

### Step 1: Understanding the Linear Congruence

A linear congruence of the form $ ax \equiv b \pmod{m} $ has a solution **if and only if** $ \gcd(a, m) \mid b $.

In our case:
- $ a = 8 $
- $ b = 1 $
- $ m = p $, where $ p $ is a **prime number**

So, the congruence $ 8x \equiv 1 \pmod{p} $ has a solution **if and only if** $ \gcd(8, p) \mid 1 $.

---

### Step 2: Analyzing the GCD Condition

Since $ p $ is a prime number, the possible values of $ \gcd(8, p) $ are:

- $ \gcd(8, p) = 1 $, if $ p \nmid 8 $
- $ \gcd(8, p) = p $, if $ p \mid 8 $

Now, the condition $ \gcd(8, p) \mid 1 $ is only satisfied when $ \gcd(8, p) = 1 $, because the only positive integer that divides 1 is 1 itself.

So, the congruence has **no solution** if and only if $ \gcd(8, p) \neq 1 $, i.e., when $ p \mid 8 $.

---

### Step 3: Identifying Primes Dividing 8

The prime factorization of 8 is:

$$
8 = 2^3
$$

So, the only prime that divides 8 is $ p = 2 $.

Thus, the only prime $ p $ for which $ \gcd(8, p) \neq 1 $ is $ p = 2 $.

---

### Step 4: Verifying the Case $ p = 2 $

Let’s check whether the congruence $ 8x \equiv 1 \pmod{2} $ has a solution.

- $ 8 \equiv 0 \pmod{2} $, so the congruence becomes:
  $$
  0 \cdot x \equiv 1 \pmod{2} \Rightarrow 0 \equiv 1 \pmod{2}
  $$
  This is a contradiction, so there is **no solution**.

Therefore, $ p = 2 $ is the **only** prime for which the congruence has no solution.

---

### Step 5: Summing the Primes

Since there is only one such prime, $ p = 2 $, the sum of all such primes is:

$$
\boxed{2}
$$

[Assistant]
Explanation:
Okay, so I need to figure out for which prime numbers p the congruence 8x ≡ 1 mod p has no solutions. Then, sum all those primes. Hmm, let me start by recalling some number theory concepts.

First, the congruence 8x ≡ 1 mod p is a linear congruence. I remember that a linear congruence ax ≡ b mod m has a solution if and only if gcd(a, m) divides b. In this case, a is 8, b is 1, and m is p, which is a prime. So, the condition for the congruence to have a solution is that gcd(8, p) divides 1.

But since p is a prime, let's think about what gcd(8, p) can be. The gcd of 8 and p can only be 1 or p, right? Because p is prime. If p divides 8, then gcd(8, p) is p. Otherwise, it's 1.

So, if p divides 8, then gcd(8, p) = p. But since p divides 8, p must be one of the prime factors of 8. The prime factors of 8 are just 2, since 8 is 2^3. Therefore, the only prime p for which gcd(8, p) is not 1 is p = 2.

So, for p ≠ 2, gcd(8, p) = 1, which divides 1, so the congruence 8x ≡ 1 mod p has a solution. However, for p = 2, gcd(8, 2) = 2. Then, since 2 does not divide 1 (because 1 divided by 2 is not an integer), the congruence 8x ≡ 1 mod 2 has no solution.

Wait, let me check that. Let me test p = 2. The congruence is 8x ≡ 1 mod 2. But 8 mod 2 is 0, so the equation becomes 0x ≡ 1 mod 2, which simplifies to 0 ≡ 1 mod 2. That's obviously false, so there's no solution. So p = 2 is such a prime.

But the question says "finitely many primes p" for which the congruence has no solutions. But wait, primes other than 2, the congruence has solutions? Let me check with another prime, say p = 3. Then 8x ≡ 1 mod 3. Since 8 mod 3 is 2, so 2x ≡ 1 mod 3. Then x ≡ 2 mod 3, since 2*2=4≡1 mod 3. So that works. Similarly, for p = 5: 8x ≡ 1 mod 5. 8 mod 5 is 3, so 3x ≡ 1 mod 5. Multiply both sides by inverse of 3 mod 5. The inverse of 3 mod 5 is 2 because 3*2=6≡1 mod 5. So x ≡ 2 mod 5. So that works.

So seems like only p=2 is the prime where there's no solution. But wait, the problem says "finitely many primes p", which is true because there's only one such prime. But let me check another prime, like p=7. 8x ≡1 mod 7. 8 mod 7 is 1, so 1x ≡1 mod 7, so x≡1 mod 7. So that works.

Wait, so maybe only p=2? But let me think again. The original question says "finitely many primes p", so maybe there are more? Wait, but according to the logic above, only p=2 divides 8, so only p=2. But maybe I made a mistake here?

Wait, let me think again. The congruence 8x ≡1 mod p has no solution if and only if gcd(8, p) does not divide 1. Since gcd(8, p) is either 1 or p. So, if gcd(8, p) divides 1, then since gcd(8, p) is either 1 or p, then p divides 1? Wait, no. Wait, the condition is that gcd(8, p) divides 1. So, gcd(8, p) must be a divisor of 1. The only positive integer that divides 1 is 1 itself. Therefore, gcd(8, p) must be 1. Therefore, the congruence has a solution if and only if gcd(8, p) = 1. Therefore, the congruence has no solution if and only if gcd(8, p) ≠ 1. Since p is prime, gcd(8, p) ≠ 1 only when p divides 8. Therefore, p divides 8. The primes dividing 8 are only 2. Therefore, only p=2. Therefore, the only prime is 2. Therefore, the sum of all such primes is 2.

But wait, the problem says "finitely many primes p", which is true, since there's only one. But maybe I need to check if there are other primes where 8x ≡1 mod p has no solution? Let me think again. Suppose p divides 8, so p=2. Then, as we saw, there's no solution. For other primes, since gcd(8, p)=1, so there is a solution. Therefore, only p=2. Therefore, the answer is 2.

But wait, let me check p=2 again. The equation is 8x ≡1 mod 2. Since 8 is congruent to 0 mod 2, so 0x ≡1 mod 2, which is 0≡1 mod 2, which is impossible. Therefore, no solution. So p=2 is the only prime. Therefore, the sum is 2.

But the problem says "finitely many primes p", which is true, but maybe I need to check if there are other primes? For example, suppose p divides 8? But 8 is 2^3, so only prime divisor is 2. Therefore, only p=2.

Wait, but maybe there's another way to think about this? For example, the equation 8x ≡1 mod p has a solution if and only if 8 is invertible modulo p. Which is equivalent to gcd(8, p)=1. Therefore, 8 is invertible modulo p if and only if p does not divide 8. Therefore, the equation has no solution if and only if p divides 8. Therefore, again, only p=2.

Therefore, the answer is 2. Therefore, the sum is 2.

But wait, the problem says "the sum of all such p". So if there is only one such prime, which is 2, then the sum is 2.

But maybe I made a mistake here? Let me check with p=2. Let me check if there is any x such that 8x ≡1 mod 2. Since 8x mod 2 is 0, because 8 is even. So 0 ≡1 mod 2? No. Therefore, no solution.

But maybe there's another prime? Let me think. Suppose p= something else. For example, p= 1? Wait, p is prime, so p=1 is not prime. So no.

Wait, maybe negative primes? No, primes are positive integers greater than 1.

Alternatively, maybe I need to think about the multiplicative inverse. The equation 8x ≡1 mod p has a solution if and only if 8 has a multiplicative inverse modulo p. Which is true if and only if 8 and p are coprime. Which is true if and only if p is not a divisor of 8. Since p is prime, p divides 8 only if p=2. Therefore, only p=2.

Therefore, the answer is 2. Therefore, the sum is 2.

But maybe the problem is trickier? Let me check again.

Wait, maybe there's a case when p divides 8? But p is prime. So p divides 8 implies p=2. So only p=2.

Alternatively, maybe the problem is in the wording? "the congruence 8x ≡1 mod p has no solutions x". So maybe there are other primes where 8x ≡1 mod p has no solution?

Wait, suppose p divides 8x -1. But for the equation to have no solution, there is no x such that 8x ≡1 mod p. Which is equivalent to saying that 8 is not invertible mod p. Which again is equivalent to gcd(8, p) ≠1. Which again is equivalent to p divides 8. Which is p=2.

Therefore, I think my conclusion is correct. Therefore, the only prime is 2, so the sum is 2.

But maybe the problem is expecting more primes? Let me check with p=  something else. For example, p= 17. Let me check if 8x ≡1 mod 17 has a solution. Let me compute 8^{-1} mod 17. Since 8*2=16≡-1 mod17, so 8*(-2)≡1 mod17. Therefore, -2 mod17 is 15. Therefore, x≡15 mod17. Therefore, solution exists.

Another prime, p=7. 8x≡1 mod7. 8≡1 mod7, so x≡1 mod7.

p=3: 8≡2 mod3, so 2x≡1 mod3. x≡2 mod3.

p=5: 8≡3 mod5. 3x≡1 mod5. x≡2 mod5.

p=11: 8x≡1 mod11. Let me compute inverse of 8 mod11. 8*? ≡1 mod11. 8* 7=56≡1 mod11 (since 55 is 5*11). So x≡7 mod11.

So all these primes have solutions.

So only p=2.

Therefore, the answer is 2.

But the problem says "finitely many primes p", which is true, since there's only one. Therefore, the sum is 2.

But maybe I need to check if there are other primes where 8x ≡1 mod p has no solution? For example, suppose p divides 8x -1. But for there to be no solution, there is no x such that 8x ≡1 mod p. Which would mean that 8x -1 is never divisible by p. Which would mean that 8x ≡1 mod p is impossible. Which again is equivalent to 8 not being invertible mod p. Which again is equivalent to p divides 8.

Therefore, only p=2.

Therefore, the answer is 2.

But maybe the problem is in the way the question is phrased? Let me check again.

"Determine the sum of all such p." So if there is only one such prime, which is 2, then the answer is 2.

Alternatively, maybe I made a mistake in thinking that p divides 8. Let me think again.

The condition for the congruence 8x ≡1 mod p to have no solution is that gcd(8, p) does not divide 1. Since gcd(8, p) divides 1 only if gcd(8, p)=1. Therefore, if gcd(8, p) ≠1, then the congruence has no solution. Since p is prime, gcd(8, p) is p if p divides 8, else 1. Therefore, p divides 8 implies p=2. Therefore, only p=2.

Therefore, the answer is 2.

But maybe the problem is expecting more? Wait, maybe I need to check if p divides 8x -1? But that's not the case. The congruence 8x ≡1 mod p has no solution if and only if there is no x such that 8x -1 is divisible by p. Which is equivalent to saying that 8x ≡1 mod p is impossible. Which is equivalent to 8 not being invertible mod p. Which is equivalent to gcd(8, p) ≠1. Which again is equivalent to p divides 8.

Therefore, only p=2.

Therefore, the answer is 2.

But maybe the problem is in the original question? Let me check again.

Original question: "There are finitely many primes p for which the congruence 8x≡1 mod p has no solutions x. Determine the sum of all such p."

Yes, so only p=2. Therefore, sum is 2.

But maybe I need to check if p=2 is considered? Since 2 is a prime. Yes.

Alternatively, maybe there's a mistake in my logic? Let me think again.

Suppose p is a prime. The equation 8x ≡1 mod p.

If p ≠2, then 8 and p are coprime, since p is odd. Therefore, 8 has an inverse modulo p, so there is a solution.

If p=2, then 8x ≡1 mod 2. Since 8 is even, 8x is 0 mod 2, so 0 ≡1 mod 2, which is false. Therefore, no solution.

Therefore, only p=2.

Therefore, the answer is 2.

But maybe the problem is expecting more? Let me check with p=  something else. For example, p= 8? But p must be prime. 8 is not prime.

Alternatively, maybe the problem is in the case when p divides 1? But p divides 1 only if p=1, which is not prime.

Therefore, no.

Therefore, I think my conclusion is correct. The only prime is 2, so the sum is 2.

But maybe the problem is from somewhere, and the answer is different? Wait, maybe I need to check if there are other primes where 8x ≡1 mod p has no solution.

Wait, suppose p divides 8x -1. For there to be no solution, there is no x such that p divides 8x -1. Which would mean that 8x ≡1 mod p is impossible. Which is equivalent to 8 not being invertible mod p. Which again is equivalent to gcd(8, p) ≠1. Which again is equivalent to p divides 8.

Therefore, only p=2.

Therefore, the answer is 2.

Therefore, I think the answer is 2.

**Final Answer**
The sum of all such primes is \boxed{2}.

To determine the sum of all prime numbers $ p $ for which the congruence

$$
8x \equiv 1 \pmod{p}
$$

has **no solutions**, we begin by analyzing the conditions under which such a linear congruence has a solution.

---

### Step 1: Understanding the Linear Congruence

A linear congruence of the form $ ax \equiv b \pmod{m} $ has a solution **if and only if** $ \gcd(a, m) \mid b $.

In our case:
- $ a = 8 $
- $ b = 1 $
- $ m = p $, where $ p $ is a **prime number**

So, the congruence $ 8x \equiv 1 \pmod{p} $ has a solution **if and only if** $ \gcd(8, p) \mid 1 $.

---

### Step 2: Analyzing the GCD Condition

Since $ p $ is a prime number, the possible values of $ \gcd(8, p) $ are:

- $ \gcd(8, p) = 1 $, if $ p \nmid 8 $
- $ \gcd(8, p) = p $, if $ p \mid 8 $

Now, the condition $ \gcd(8, p) \mid 1 $ is only satisfied when $ \gcd(8, p) = 1 $, because the only positive integer that divides 1 is 1 itself.

So, the congruence has **no solution** if and only if $ \gcd(8, p) \neq 1 $, i.e., when $ p \mid 8 $.

---

### Step 3: Identifying Primes Dividing 8

The prime factorization of 8 is:

$$
8 = 2^3
$$

So, the only prime that divides 8 is $ p = 2 $.

Thus, the only prime $ p $ for which $ \gcd(8, p) \neq 1 $ is $ p = 2 $.

---

### Step 4: Verifying the Case $ p = 2 $

Let’s check whether the congruence $ 8x \equiv 1 \pmod{2} $ has a solution.

- $ 8 \equiv 0 \pmod{2} $, so the congruence becomes:
  $$
  0 \cdot x \equiv 1 \pmod{2} \Rightarrow 0 \equiv 1 \pmod{2}
  $$
  This is a contradiction, so there is **no solution**.

Therefore, $ p = 2 $ is the **only** prime for which the congruence has no solution.

---

### Step 5: Summing the Primes

Since there is only one such prime, $ p = 2 $, the sum of all such primes is:

$$
\boxed{2}
$$

Solution: A solution exists if and only if $8$ is invertible modulo $p$. In other words, $\gcd(8,p)=1$. Since $8=2^3$ is a power of $2$, $8$ is invertible modulo $q$ if and only if $q$ is an odd integer. All primes except for $2$ are odd, so the number we are looking for is $\boxed{2}$.


Explanation:
Okay, so I need to simplify this big fraction where the numerator is the product of terms like (2^4 + 1/4), (4^4 + 1/4), all the way up to (2n)^4 + 1/4, and the denominator is the product of similar terms but with odd numbers: (1^4 + 1/4), (3^4 + 1/4), ..., up to (2n - 1)^4 + 1/4.

First, let me write down the expression more clearly:

$$
\frac{(2^4 + \frac{1}{4})(4^4 + \frac{1}{4}) \dotsm [(2n)^4 + \frac{1}{4}]}{(1^4 + \frac{1}{4})(3^4 + \frac{1}{4}) \dotsm [(2n - 1)^4 + \frac{1}{4}]}
$$

So, the numerator is the product of (2k)^4 + 1/4 for k from 1 to n, and the denominator is the product of (2k - 1)^4 + 1/4 for k from 1 to n.

Hmm, maybe I can first try to simplify each term in the numerator and denominator. Let me take a general term, say, for a number x, x^4 + 1/4. Maybe there's a way to factor this expression?

I remember that x^4 + 4y^4 can be factored using the Sophie Germain identity: x^4 + 4y^4 = (x^2 + 2xy + 2y^2)(x^2 - 2xy + 2y^2). Let me check if that applies here.

In our case, the term is x^4 + 1/4. Let me write 1/4 as 4*(1/4)^2? Wait, 4*(1/4)^2 is 4*(1/16) = 1/4. So yes! So if I set y = 1/2, then x^4 + 4*(1/2)^4 = x^4 + 1/4. Therefore, applying the Sophie Germain identity:

x^4 + 1/4 = (x^2 + 2x*(1/2) + 2*(1/2)^2)(x^2 - 2x*(1/2) + 2*(1/2)^2)

Simplify each part:

First factor: x^2 + x + 2*(1/4) = x^2 + x + 1/2

Second factor: x^2 - x + 1/2

So, x^4 + 1/4 factors into (x^2 + x + 1/2)(x^2 - x + 1/2).

Therefore, each term in the numerator and denominator can be factored like this. Let me write that down for both numerator and denominator terms.

For the numerator terms, which are (2k)^4 + 1/4, substituting x = 2k:

( (2k)^2 + 2k + 1/2 ) * ( (2k)^2 - 2k + 1/2 )

Similarly, for the denominator terms, which are (2k - 1)^4 + 1/4, substituting x = 2k - 1:

( (2k - 1)^2 + (2k - 1) + 1/2 ) * ( (2k - 1)^2 - (2k - 1) + 1/2 )

So, if I can factor each term in numerator and denominator, maybe there will be some cancellation when I take the product over all terms.

Let me compute these factors for both numerator and denominator.

Starting with the numerator terms:

First factor for numerator term: ( (2k)^2 + 2k + 1/2 ) = 4k^2 + 2k + 1/2

Second factor: ( (2k)^2 - 2k + 1/2 ) = 4k^2 - 2k + 1/2

Similarly, for the denominator terms:

First factor: ( (2k - 1)^2 + (2k - 1) + 1/2 )

Let me compute that:

(2k - 1)^2 = 4k^2 - 4k + 1

Adding (2k - 1): 4k^2 - 4k + 1 + 2k - 1 = 4k^2 - 2k

Adding 1/2: 4k^2 - 2k + 1/2

Wait, so the first factor for the denominator term is 4k^2 - 2k + 1/2?

Wait, let me check again:

Original expression: (x^2 + x + 1/2) where x = 2k - 1.

So x^2 is (2k - 1)^2 = 4k² - 4k + 1

x is 2k - 1

So adding x^2 + x + 1/2:

(4k² - 4k + 1) + (2k - 1) + 1/2 = 4k² - 4k + 1 + 2k - 1 + 1/2 = 4k² - 2k + 0 + 1/2 = 4k² - 2k + 1/2. Yes, same as the second factor in the numerator term.

Similarly, the second factor for the denominator term is (x² - x + 1/2) where x = 2k - 1.

So x² - x + 1/2 = (4k² - 4k + 1) - (2k - 1) + 1/2 = 4k² - 4k + 1 - 2k + 1 + 1/2 = 4k² - 6k + 2 + 1/2 = 4k² - 6k + 5/2? Wait, let me check again:

Wait, step by step:

x² - x + 1/2 = (4k² - 4k + 1) - (2k - 1) + 1/2

First, expand the subtraction: 4k² - 4k + 1 - 2k + 1 + 1/2

Combine like terms:

4k² - 4k - 2k + 1 + 1 + 1/2 = 4k² - 6k + 2 + 1/2 = 4k² - 6k + 5/2.

Hmm, so the second factor for the denominator term is 4k² - 6k + 5/2.

Wait, but the first factor for the denominator term is 4k² - 2k + 1/2, which is same as the second factor in the numerator term.

So, if I look at the numerator terms, each term factors into (4k² + 2k + 1/2)(4k² - 2k + 1/2).

Similarly, each denominator term factors into (4k² - 2k + 1/2)(4k² - 6k + 5/2).

Wait, so if I write the entire product for numerator and denominator, maybe there is some telescoping?

Let me think. Let me denote for each term in the numerator:

Term_numerator_k = (A_k)(B_k), where A_k = 4k² + 2k + 1/2 and B_k = 4k² - 2k + 1/2.

Similarly, Term_denominator_k = (C_k)(D_k), where C_k = 4k² - 2k + 1/2 and D_k = 4k² - 6k + 5/2.

Therefore, the entire fraction is [Product_{k=1}^n (A_k B_k)] / [Product_{k=1}^n (C_k D_k)] = [Product_{k=1}^n A_k * Product_{k=1}^n B_k] / [Product_{k=1}^n C_k * Product_{k=1}^n D_k]

But notice that B_k = C_k for each k? Wait, no. Wait, for the numerator term, B_k is 4k² - 2k + 1/2, which is same as C_k for denominator term? Wait, no. Wait, for the denominator term, C_k is 4k² - 2k + 1/2. So yes, B_k = C_k. Therefore, the product of B_k from k=1 to n is same as product of C_k from k=1 to n. Therefore, the fraction simplifies to [Product A_k * Product B_k] / [Product C_k * Product D_k] = [Product A_k] / [Product D_k], since Product B_k / Product C_k = 1.

Therefore, the entire expression simplifies to [Product_{k=1}^n A_k] / [Product_{k=1}^n D_k], where A_k = 4k² + 2k + 1/2 and D_k = 4k² - 6k + 5/2.

So now, I need to compute the product of A_k divided by the product of D_k. Let me write down A_k and D_k:

A_k = 4k² + 2k + 1/2. Let me write this as fractions to make it easier. 4k² is 8k²/2, 2k is 4k/2, so adding 1/2:

A_k = (8k² + 4k + 1)/2

Similarly, D_k = 4k² - 6k + 5/2. Let me convert to fractions:

4k² is 8k²/2, -6k is -12k/2, and 5/2 is 5/2. So:

D_k = (8k² - 12k + 5)/2

Therefore, the entire expression becomes:

[Product_{k=1}^n (8k² + 4k + 1)/2 ] / [Product_{k=1}^n (8k² - 12k + 5)/2 ]

Which simplifies to [Product_{k=1}^n (8k² + 4k + 1) / 2^n ] / [Product_{k=1}^n (8k² - 12k + 5)/2^n ]

Since each term in the numerator has a denominator of 2, and there are n terms, so total denominator is 2^n. Similarly for the denominator. Therefore, dividing them, the 2^n cancels out:

[Product_{k=1}^n (8k² + 4k + 1)] / [Product_{k=1}^n (8k² - 12k + 5)]

So now, the problem reduces to simplifying this ratio of products. Let me look at the polynomials 8k² + 4k + 1 and 8k² - 12k + 5. Maybe they can be related to each other in some way, perhaps by shifting the index?

Let me check if 8k² - 12k + 5 can be expressed in terms of some expression involving (k - m) for some m. Let me try to see if there's a relationship between the numerator polynomial and denominator polynomial.

Let me consider the denominator polynomial: 8k² - 12k + 5. Let me try to write this as 8(k - a)^2 + b or something similar. Alternatively, maybe it's related to the numerator polynomial evaluated at a different k?

Alternatively, let me check if 8k² - 12k + 5 is equal to 8(k - something)^2 + something else. Let me compute discriminant or try to factor it.

Alternatively, maybe notice that 8k² - 12k + 5 can be written as 8(k - 3/2)^2 - something? Let me try:

Let me complete the square for 8k² - 12k + 5.

Factor out 8 from the first two terms:

8(k² - (12/8)k) + 5 = 8(k² - (3/2)k) + 5

Complete the square inside the parentheses:

k² - (3/2)k + ( (3/4)^2 ) - ( (3/4)^2 ) = [k - 3/4]^2 - 9/16

Therefore:

8([k - 3/4]^2 - 9/16) + 5 = 8[k - 3/4]^2 - 9/2 + 5 = 8[k - 3/4]^2 - 9/2 + 10/2 = 8[k - 3/4]^2 + 1/2

Hmm, not sure if that helps. Let me check the numerator polynomial: 8k² + 4k + 1. Let me try completing the square here as well.

Factor out 8 from first two terms:

8(k² + (4/8)k) + 1 = 8(k² + (1/2)k) + 1

Complete the square:

k² + (1/2)k + (1/4)^2 - (1/4)^2 = [k + 1/4]^2 - 1/16

Therefore:

8([k + 1/4]^2 - 1/16) + 1 = 8[k + 1/4]^2 - 8*(1/16) + 1 = 8[k + 1/4]^2 - 1/2 + 1 = 8[k + 1/4]^2 + 1/2

So both numerator and denominator polynomials can be written as 8*(something)^2 + 1/2. Not sure if that helps yet.

Alternatively, maybe there's a relationship between the numerator polynomial evaluated at k and the denominator polynomial evaluated at k + something?

Let me check for specific values. Let me take k and see if denominator polynomial at k is related to numerator polynomial at some other value.

Let me compute denominator polynomial at k: 8k² - 12k + 5. Let me see if this is equal to numerator polynomial evaluated at k - m for some m.

Suppose I try m = 1. Let me compute numerator polynomial at k - 1:

8(k - 1)^2 + 4(k - 1) + 1 = 8(k² - 2k + 1) + 4k - 4 + 1 = 8k² - 16k + 8 + 4k - 4 + 1 = 8k² - 12k + 5.

Oh! That's exactly the denominator polynomial!

So, 8(k - 1)^2 + 4(k - 1) + 1 = 8k² - 12k + 5.

Therefore, D_k = A_{k - 1} where A_{k} = 8k² + 4k + 1.

Therefore, the denominator polynomial D_k is equal to the numerator polynomial A_{k - 1}.

Therefore, the product in the denominator is Product_{k=1}^n D_k = Product_{k=1}^n A_{k - 1} = Product_{m=0}^{n - 1} A_m, where m = k - 1.

Therefore, the entire expression becomes [Product_{k=1}^n A_k] / [Product_{m=0}^{n - 1} A_m] = A_n / A_0.

Because when you divide the product from k=1 to n by the product from m=0 to n-1, all the terms from A_1 to A_{n-1} cancel out, leaving A_n / A_0.

So, the entire expression simplifies to A_n / A_0.

Now, let me compute A_n and A_0.

Recall that A_k = 8k² + 4k + 1.

Therefore, A_n = 8n² + 4n + 1.

A_0 = 8*(0)^2 + 4*0 + 1 = 0 + 0 + 1 = 1.

Therefore, the entire expression simplifies to (8n² + 4n + 1)/1 = 8n² + 4n + 1.

Wait, so is that the final answer? Let me check with small n to verify.

Let me test n = 1.

Original expression: numerator is (2^4 + 1/4) = 16 + 1/4 = 65/4. Denominator is (1^4 + 1/4) = 1 + 1/4 = 5/4. Therefore, the ratio is (65/4) / (5/4) = 65/5 = 13.

Now, according to the formula 8n² + 4n + 1, when n = 1: 8 + 4 + 1 = 13. Correct.

Let me check n = 2.

Original expression: numerator is (2^4 + 1/4)(4^4 + 1/4) = (16 + 1/4)(256 + 1/4) = (65/4)(1025/4) = (65 * 1025)/16.

Denominator is (1^4 + 1/4)(3^4 + 1/4) = (5/4)(81 + 1/4) = (5/4)(325/4) = (5 * 325)/16.

Therefore, the ratio is (65 * 1025) / (5 * 325) = (13 * 5 * 1025) / (5 * 65) )? Wait, let me compute step by step:

First, 65 / 5 = 13. Then 1025 / 325. Let me compute 1025 / 325. 325 * 3 = 975, so 1025 - 975 = 50. So 3 + 50/325 = 3 + 10/65 = 3 + 2/13 = 3.1538... But maybe better to factor:

1025 = 5 * 205 = 5 * 5 * 41 = 25 * 41

325 = 25 * 13

Therefore, 1025 / 325 = (25 * 41) / (25 * 13) ) = 41 / 13.

Therefore, the ratio is 13 * (41 / 13) = 41.

Now, according to the formula 8n² + 4n + 1 with n = 2: 8*(4) + 4*2 + 1 = 32 + 8 + 1 = 41. Correct.

Another check with n = 3.

Original expression numerator: (2^4 + 1/4)(4^4 + 1/4)(6^4 + 1/4) = (65/4)(1025/4)(1296 + 1/4) = (65/4)(1025/4)(5185/4).

Denominator: (1^4 + 1/4)(3^4 + 1/4)(5^4 + 1/4) = (5/4)(325/4)(625 + 1/4) = (5/4)(325/4)(2501/4).

So the ratio is [65 * 1025 * 5185] / [5 * 325 * 2501] * (1/4^3) / (1/4^3) = same as [65 * 1025 * 5185] / [5 * 325 * 2501].

Let me compute step by step:

First, 65 / 5 = 13.

Then, 1025 / 325 = 41 / 13 as before.

Then, 5185 / 2501. Let me check if 2501 * 2 = 5002, so 5185 - 5002 = 183. Not sure. Let me check if 2501 is a square? 50^2 = 2500, so 50^2 + 1 = 2501. So sqrt(2501) is irrational. Wait, but maybe 5185 is related?

Wait, 5185 divided by 2501. Let me check if 2501 * 2 = 5002, 5185 - 5002 = 183. 2501 * 2 + 183 = 5185. Not sure. Alternatively, maybe there's a pattern here.

But according to our formula, the result should be 8*(3)^2 + 4*3 + 1 = 8*9 + 12 + 1 = 72 + 13 = 85.

So let me compute the ratio:

[13 * (41/13) * (5185 / 2501)] = 41 * (5185 / 2501).

So need to check if 5185 / 2501 = 85 / something? Wait, 2501 * 2 = 5002, 2501 * 2.07... Let me check if 5185 = 2501 * something.

Alternatively, maybe 5185 = 8*648 + 1? Wait, 8*648 = 5184, so 5185 = 8*648 + 1.

But 2501 is 50^2 + 1. Not sure. Alternatively, maybe there's a relation between 5185 and 2501. Let me check if 5185 = 2501 * 2 + 183, as before.

Alternatively, maybe I made a mistake in the calculation. Let me think again.

Alternatively, since according to our formula, the answer should be 8n² + 4n + 1 = 8*9 + 12 + 1 = 85. Therefore, the ratio should be 85.

So let me compute 41 * (5185 / 2501). Let me compute 5185 divided by 2501. Let me note that 2501 * 2 = 5002, so 5185 - 5002 = 183. Therefore, 5185 / 2501 = 2 + 183/2501.

But 183 is 3*61, and 2501 is 50^2 + 1. Wait, 50^2 is 2500, so 2501 is 50^2 + 1. Not sure.

Alternatively, maybe there's a mistake in my previous steps? Let me check again.

Wait, if the formula is correct, then the ratio should be 85. Let me check if 41 * (5185 / 2501) equals 85.

Let me compute 41 * 5185 = ?

Well, 40 * 5185 = 207,400

Plus 1 * 5185 = 5,185

Total: 207,400 + 5,185 = 212,585

Now, divide by 2501: 212,585 / 2501. Let me check how many times 2501 goes into 212,585.

Well, 2501 * 80 = 200,080

Subtract: 212,585 - 200,080 = 12,505

Now, 2501 * 5 = 12,505.

Therefore, total is 80 + 5 = 85.

Yes! Therefore, 212,585 / 2501 = 85. Therefore, the ratio is indeed 85. Which matches the formula.

Therefore, the formula seems to hold.

Therefore, the simplified expression is 8n² + 4n + 1.

But let me check if this can be simplified further or written in another form.

Alternatively, maybe factor it? Let me see:

8n² + 4n + 1. Let me check discriminant: 16 - 32 = -16. So it doesn't factor over real numbers. So probably this is the simplest form.

Alternatively, maybe write it as (something)^2 + something? Not sure.

Alternatively, maybe notice that 8n² + 4n + 1 = (2n + 1)^2 + 2n^2? Let me check:

(2n + 1)^2 = 4n² + 4n + 1. Then adding 2n² gives 6n² + 4n + 1, which is not.

Alternatively, maybe 8n² + 4n + 1 = 4*(2n² + n) + 1. Not sure.

Alternatively, maybe write as 8n² + 4n + 1 = 8n² + 4n + 1.

Alternatively, maybe express in terms of (2n)^2 + something?

But since the problem says "simplify", and 8n² + 4n + 1 is a quadratic, and it's already simplified, so probably that is the answer.

But let me check again the steps to make sure I didn't make a mistake.

We started with the original expression, factored each term using Sophie Germain identity, noticed that the denominator terms had factors that canceled with numerator terms, leading to telescoping product, which resulted in A_n / A_0, which is 8n² + 4n + 1.

Yes, and verified with n=1, n=2, n=3. So seems correct.

Therefore, the simplified expression is 8n² + 4n + 1.

But let me check if there's another way to write this. For example, maybe in terms of (something else).

Alternatively, note that 8n² + 4n + 1 = 4*(2n² + n) + 1. Not sure.

Alternatively, maybe write as (2n + 1)^2 + 2n²? Wait:

(2n + 1)^2 = 4n² + 4n + 1. Then adding 2n² gives 6n² + 4n + 1. Not helpful.

Alternatively, maybe think of 8n² + 4n + 1 as (something else).

Alternatively, maybe write as 8n² + 4n + 1 = (something)^2 + ... but not sure.

Alternatively, maybe check if it's equal to (something related to the original terms).

But since the problem says "simplify", and 8n² + 4n + 1 is a simplified polynomial, I think that is the answer.

Therefore, the final answer is 8n² + 4n + 1, which can be written as \boxed{8n^2 + 4n + 1}.

**Final Answer**
\boxed{8n^2 + 4n + 1}

To simplify the expression

$$
\frac{(2^4 + \frac{1}{4})(4^4 + \frac{1}{4}) \dotsm [(2n)^4 + \frac{1}{4}]}{(1^4 + \frac{1}{4})(3^4 + \frac{1}{4}) \dotsm [(2n - 1)^4 + \frac{1}{4}]}
$$

we begin by recognizing a pattern in the terms. Each term in both the numerator and the denominator is of the form $ x^4 + \frac{1}{4} $, where $ x $ is an even or odd integer, respectively.

---

### Step 1: Factor the General Term

We use the **Sophie Germain identity**:

$$
x^4 + 4y^4 = (x^2 + 2xy + 2y^2)(x^2 - 2xy + 2y^2)
$$

Set $ y = \frac{1}{2} $, so that $ 4y^4 = \frac{1}{4} $. Then:

$$
x^4 + \frac{1}{4} = (x^2 + x + \frac{1}{2})(x^2 - x + \frac{1}{2})
$$

This allows us to factor each term in the numerator and denominator.

---

### Step 2: Apply the Factorization

Let’s denote:

- Numerator terms: $ (2k)^4 + \frac{1}{4} = (4k^2 + 2k + \frac{1}{2})(4k^2 - 2k + \frac{1}{2}) $
- Denominator terms: $ (2k - 1)^4 + \frac{1}{4} = (4k^2 - 2k + \frac{1}{2})(4k^2 - 6k + \frac{5}{2}) $

So the full expression becomes:

$$
\frac{\prod_{k=1}^n (4k^2 + 2k + \frac{1}{2})(4k^2 - 2k + \frac{1}{2})}{\prod_{k=1}^n (4k^2 - 2k + \frac{1}{2})(4k^2 - 6k + \frac{5}{2})}
$$

Notice that the second factor in the numerator, $ 4k^2 - 2k + \frac{1}{2} $, is the same as the first factor in the denominator. This allows for **cancellation** of the middle terms across the numerator and denominator.

After cancellation, the expression reduces to:

$$
\frac{\prod_{k=1}^n (4k^2 + 2k + \frac{1}{2})}{\prod_{k=1}^n (4k^2 - 6k + \frac{5}{2})}
$$

---

### Step 3: Express in Terms of a Single Polynomial

Let’s define:

- $ A_k = 4k^2 + 2k + \frac{1}{2} $
- $ D_k = 4k^2 - 6k + \frac{5}{2} $

We observe that:

$$
D_k = A_{k-1}
$$

This is verified by substituting $ k-1 $ into $ A_k $:

$$
A_{k-1} = 4(k-1)^2 + 2(k-1) + \frac{1}{2} = 4k^2 - 8k + 4 + 2k - 2 + \frac{1}{2} = 4k^2 - 6k + \frac{5}{2}
$$

So the denominator becomes:

$$
\prod_{k=1}^n D_k = \prod_{k=1}^n A_{k-1} = \prod_{m=0}^{n-1} A_m
$$

Thus, the entire expression becomes:

$$
\frac{\prod_{k=1}^n A_k}{\prod_{m=0}^{n-1} A_m} = \frac{A_n}{A_0}
$$

---

### Step 4: Evaluate the Final Expression

Recall:

$$
A_k = 4k^2 + 2k + \frac{1}{2}
$$

So:

- $ A_n = 4n^2 + 2n + \frac{1}{2} $
- $ A_0 = 4(0)^2 + 2(0) + \frac{1}{2} = \frac{1}{2} $

Therefore:

$$
\frac{A_n}{A_0} = \frac{4n^2 + 2n + \frac{1}{2}}{\frac{1}{2}} = 8n^2 + 4n + 1
$$

---

### Final Answer

$$
\boxed{8n^2 + 4n + 1}
$$

[Assistant]
Explanation:
Okay, so I need to simplify this big fraction where the numerator is the product of terms like (2^4 + 1/4), (4^4 + 1/4), all the way up to (2n)^4 + 1/4, and the denominator is the product of similar terms but with odd numbers: (1^4 + 1/4), (3^4 + 1/4), ..., up to (2n - 1)^4 + 1/4.

First, let me write down the expression more clearly:

$$
\frac{(2^4 + \frac{1}{4})(4^4 + \frac{1}{4}) \dotsm [(2n)^4 + \frac{1}{4}]}{(1^4 + \frac{1}{4})(3^4 + \frac{1}{4}) \dotsm [(2n - 1)^4 + \frac{1}{4}]}
$$

So, the numerator is the product of (2k)^4 + 1/4 for k from 1 to n, and the denominator is the product of (2k - 1)^4 + 1/4 for k from 1 to n.

Hmm, maybe I can first try to simplify each term in the numerator and denominator. Let me take a general term, say, for a number x, x^4 + 1/4. Maybe there's a way to factor this expression?

I remember that x^4 + 4y^4 can be factored using the Sophie Germain identity: x^4 + 4y^4 = (x^2 + 2xy + 2y^2)(x^2 - 2xy + 2y^2). Let me check if that applies here.

In our case, the term is x^4 + 1/4. Let me write 1/4 as 4*(1/4)^2? Wait, 4*(1/4)^2 is 4*(1/16) = 1/4. So yes! So if I set y = 1/2, then x^4 + 4*(1/2)^4 = x^4 + 1/4. Therefore, applying the Sophie Germain identity:

x^4 + 1/4 = (x^2 + 2x*(1/2) + 2*(1/2)^2)(x^2 - 2x*(1/2) + 2*(1/2)^2)

Simplify each part:

First factor: x^2 + x + 2*(1/4) = x^2 + x + 1/2

Second factor: x^2 - x + 1/2

So, x^4 + 1/4 factors into (x^2 + x + 1/2)(x^2 - x + 1/2).

Therefore, each term in the numerator and denominator can be factored like this. Let me write that down for both numerator and denominator terms.

For the numerator terms, which are (2k)^4 + 1/4, substituting x = 2k:

( (2k)^2 + 2k + 1/2 ) * ( (2k)^2 - 2k + 1/2 )

Similarly, for the denominator terms, which are (2k - 1)^4 + 1/4, substituting x = 2k - 1:

( (2k - 1)^2 + (2k - 1) + 1/2 ) * ( (2k - 1)^2 - (2k - 1) + 1/2 )

So, if I can factor each term in numerator and denominator, maybe there will be some cancellation when I take the product over all terms.

Let me compute these factors for both numerator and denominator.

Starting with the numerator terms:

First factor for numerator term: ( (2k)^2 + 2k + 1/2 ) = 4k^2 + 2k + 1/2

Second factor: ( (2k)^2 - 2k + 1/2 ) = 4k^2 - 2k + 1/2

Similarly, for the denominator terms:

First factor: ( (2k - 1)^2 + (2k - 1) + 1/2 )

Let me compute that:

(2k - 1)^2 = 4k^2 - 4k + 1

Adding (2k - 1): 4k^2 - 4k + 1 + 2k - 1 = 4k^2 - 2k

Adding 1/2: 4k^2 - 2k + 1/2

Wait, so the first factor for the denominator term is 4k^2 - 2k + 1/2?

Wait, let me check again:

Original expression: (x^2 + x + 1/2) where x = 2k - 1.

So x^2 is (2k - 1)^2 = 4k² - 4k + 1

x is 2k - 1

So adding x^2 + x + 1/2:

(4k² - 4k + 1) + (2k - 1) + 1/2 = 4k² - 4k + 1 + 2k - 1 + 1/2 = 4k² - 2k + 0 + 1/2 = 4k² - 2k + 1/2. Yes, same as the second factor in the numerator term.

Similarly, the second factor for the denominator term is (x² - x + 1/2) where x = 2k - 1.

So x² - x + 1/2 = (4k² - 4k + 1) - (2k - 1) + 1/2 = 4k² - 4k + 1 - 2k + 1 + 1/2 = 4k² - 6k + 2 + 1/2 = 4k² - 6k + 5/2? Wait, let me check again:

Wait, step by step:

x² - x + 1/2 = (4k² - 4k + 1) - (2k - 1) + 1/2

First, expand the subtraction: 4k² - 4k + 1 - 2k + 1 + 1/2

Combine like terms:

4k² - 4k - 2k + 1 + 1 + 1/2 = 4k² - 6k + 2 + 1/2 = 4k² - 6k + 5/2.

Hmm, so the second factor for the denominator term is 4k² - 6k + 5/2.

Wait, but the first factor for the denominator term is 4k² - 2k + 1/2, which is same as the second factor in the numerator term.

So, if I look at the numerator terms, each term factors into (4k² + 2k + 1/2)(4k² - 2k + 1/2).

Similarly, each denominator term factors into (4k² - 2k + 1/2)(4k² - 6k + 5/2).

Wait, so if I write the entire product for numerator and denominator, maybe there is some telescoping?

Let me think. Let me denote for each term in the numerator:

Term_numerator_k = (A_k)(B_k), where A_k = 4k² + 2k + 1/2 and B_k = 4k² - 2k + 1/2.

Similarly, Term_denominator_k = (C_k)(D_k), where C_k = 4k² - 2k + 1/2 and D_k = 4k² - 6k + 5/2.

Therefore, the entire fraction is [Product_{k=1}^n (A_k B_k)] / [Product_{k=1}^n (C_k D_k)] = [Product_{k=1}^n A_k * Product_{k=1}^n B_k] / [Product_{k=1}^n C_k * Product_{k=1}^n D_k]

But notice that B_k = C_k for each k? Wait, no. Wait, for the numerator term, B_k is 4k² - 2k + 1/2, which is same as C_k for denominator term? Wait, no. Wait, for the denominator term, C_k is 4k² - 2k + 1/2. So yes, B_k = C_k. Therefore, the product of B_k from k=1 to n is same as product of C_k from k=1 to n. Therefore, the fraction simplifies to [Product A_k * Product B_k] / [Product C_k * Product D_k] = [Product A_k] / [Product D_k], since Product B_k / Product C_k = 1.

Therefore, the entire expression simplifies to [Product_{k=1}^n A_k] / [Product_{k=1}^n D_k], where A_k = 4k² + 2k + 1/2 and D_k = 4k² - 6k + 5/2.

So now, I need to compute the product of A_k divided by the product of D_k. Let me write down A_k and D_k:

A_k = 4k² + 2k + 1/2. Let me write this as fractions to make it easier. 4k² is 8k²/2, 2k is 4k/2, so adding 1/2:

A_k = (8k² + 4k + 1)/2

Similarly, D_k = 4k² - 6k + 5/2. Let me convert to fractions:

4k² is 8k²/2, -6k is -12k/2, and 5/2 is 5/2. So:

D_k = (8k² - 12k + 5)/2

Therefore, the entire expression becomes:

[Product_{k=1}^n (8k² + 4k + 1)/2 ] / [Product_{k=1}^n (8k² - 12k + 5)/2 ]

Which simplifies to [Product_{k=1}^n (8k² + 4k + 1) / 2^n ] / [Product_{k=1}^n (8k² - 12k + 5)/2^n ]

Since each term in the numerator has a denominator of 2, and there are n terms, so total denominator is 2^n. Similarly for the denominator. Therefore, dividing them, the 2^n cancels out:

[Product_{k=1}^n (8k² + 4k + 1)] / [Product_{k=1}^n (8k² - 12k + 5)]

So now, the problem reduces to simplifying this ratio of products. Let me look at the polynomials 8k² + 4k + 1 and 8k² - 12k + 5. Maybe they can be related to each other in some way, perhaps by shifting the index?

Let me check if 8k² - 12k + 5 can be expressed in terms of some expression involving (k - m) for some m. Let me try to see if there's a relationship between the numerator polynomial and denominator polynomial.

Let me consider the denominator polynomial: 8k² - 12k + 5. Let me try to write this as 8(k - a)^2 + b or something similar. Alternatively, maybe it's related to the numerator polynomial evaluated at a different k?

Alternatively, let me check if 8k² - 12k + 5 is equal to 8(k - something)^2 + something else. Let me compute discriminant or try to factor it.

Alternatively, maybe notice that 8k² - 12k + 5 can be written as 8(k - 3/2)^2 - something? Let me try:

Let me complete the square for 8k² - 12k + 5.

Factor out 8 from the first two terms:

8(k² - (12/8)k) + 5 = 8(k² - (3/2)k) + 5

Complete the square inside the parentheses:

k² - (3/2)k + ( (3/4)^2 ) - ( (3/4)^2 ) = [k - 3/4]^2 - 9/16

Therefore:

8([k - 3/4]^2 - 9/16) + 5 = 8[k - 3/4]^2 - 9/2 + 5 = 8[k - 3/4]^2 - 9/2 + 10/2 = 8[k - 3/4]^2 + 1/2

Hmm, not sure if that helps. Let me check the numerator polynomial: 8k² + 4k + 1. Let me try completing the square here as well.

Factor out 8 from first two terms:

8(k² + (4/8)k) + 1 = 8(k² + (1/2)k) + 1

Complete the square:

k² + (1/2)k + (1/4)^2 - (1/4)^2 = [k + 1/4]^2 - 1/16

Therefore:

8([k + 1/4]^2 - 1/16) + 1 = 8[k + 1/4]^2 - 8*(1/16) + 1 = 8[k + 1/4]^2 - 1/2 + 1 = 8[k + 1/4]^2 + 1/2

So both numerator and denominator polynomials can be written as 8*(something)^2 + 1/2. Not sure if that helps yet.

Alternatively, maybe there's a relationship between the numerator polynomial evaluated at k and the denominator polynomial evaluated at k + something?

Let me check for specific values. Let me take k and see if denominator polynomial at k is related to numerator polynomial at some other value.

Let me compute denominator polynomial at k: 8k² - 12k + 5. Let me see if this is equal to numerator polynomial evaluated at k - m for some m.

Suppose I try m = 1. Let me compute numerator polynomial at k - 1:

8(k - 1)^2 + 4(k - 1) + 1 = 8(k² - 2k + 1) + 4k - 4 + 1 = 8k² - 16k + 8 + 4k - 4 + 1 = 8k² - 12k + 5.

Oh! That's exactly the denominator polynomial!

So, 8(k - 1)^2 + 4(k - 1) + 1 = 8k² - 12k + 5.

Therefore, D_k = A_{k - 1} where A_{k} = 8k² + 4k + 1.

Therefore, the denominator polynomial D_k is equal to the numerator polynomial A_{k - 1}.

Therefore, the product in the denominator is Product_{k=1}^n D_k = Product_{k=1}^n A_{k - 1} = Product_{m=0}^{n - 1} A_m, where m = k - 1.

Therefore, the entire expression becomes [Product_{k=1}^n A_k] / [Product_{m=0}^{n - 1} A_m] = A_n / A_0.

Because when you divide the product from k=1 to n by the product from m=0 to n-1, all the terms from A_1 to A_{n-1} cancel out, leaving A_n / A_0.

So, the entire expression simplifies to A_n / A_0.

Now, let me compute A_n and A_0.

Recall that A_k = 8k² + 4k + 1.

Therefore, A_n = 8n² + 4n + 1.

A_0 = 8*(0)^2 + 4*0 + 1 = 0 + 0 + 1 = 1.

Therefore, the entire expression simplifies to (8n² + 4n + 1)/1 = 8n² + 4n + 1.

Wait, so is that the final answer? Let me check with small n to verify.

Let me test n = 1.

Original expression: numerator is (2^4 + 1/4) = 16 + 1/4 = 65/4. Denominator is (1^4 + 1/4) = 1 + 1/4 = 5/4. Therefore, the ratio is (65/4) / (5/4) = 65/5 = 13.

Now, according to the formula 8n² + 4n + 1, when n = 1: 8 + 4 + 1 = 13. Correct.

Let me check n = 2.

Original expression: numerator is (2^4 + 1/4)(4^4 + 1/4) = (16 + 1/4)(256 + 1/4) = (65/4)(1025/4) = (65 * 1025)/16.

Denominator is (1^4 + 1/4)(3^4 + 1/4) = (5/4)(81 + 1/4) = (5/4)(325/4) = (5 * 325)/16.

Therefore, the ratio is (65 * 1025) / (5 * 325) = (13 * 5 * 1025) / (5 * 65) )? Wait, let me compute step by step:

First, 65 / 5 = 13. Then 1025 / 325. Let me compute 1025 / 325. 325 * 3 = 975, so 1025 - 975 = 50. So 3 + 50/325 = 3 + 10/65 = 3 + 2/13 = 3.1538... But maybe better to factor:

1025 = 5 * 205 = 5 * 5 * 41 = 25 * 41

325 = 25 * 13

Therefore, 1025 / 325 = (25 * 41) / (25 * 13) ) = 41 / 13.

Therefore, the ratio is 13 * (41 / 13) = 41.

Now, according to the formula 8n² + 4n + 1 with n = 2: 8*(4) + 4*2 + 1 = 32 + 8 + 1 = 41. Correct.

Another check with n = 3.

Original expression numerator: (2^4 + 1/4)(4^4 + 1/4)(6^4 + 1/4) = (65/4)(1025/4)(1296 + 1/4) = (65/4)(1025/4)(5185/4).

Denominator: (1^4 + 1/4)(3^4 + 1/4)(5^4 + 1/4) = (5/4)(325/4)(625 + 1/4) = (5/4)(325/4)(2501/4).

So the ratio is [65 * 1025 * 5185] / [5 * 325 * 2501] * (1/4^3) / (1/4^3) = same as [65 * 1025 * 5185] / [5 * 325 * 2501].

Let me compute step by step:

First, 65 / 5 = 13.

Then, 1025 / 325 = 41 / 13 as before.

Then, 5185 / 2501. Let me check if 2501 * 2 = 5002, so 5185 - 5002 = 183. Not sure. Let me check if 2501 is a square? 50^2 = 2500, so 50^2 + 1 = 2501. So sqrt(2501) is irrational. Wait, but maybe 5185 is related?

Wait, 5185 divided by 2501. Let me check if 2501 * 2 = 5002, 5185 - 5002 = 183. 2501 * 2 + 183 = 5185. Not sure. Alternatively, maybe there's a pattern here.

But according to our formula, the result should be 8*(3)^2 + 4*3 + 1 = 8*9 + 12 + 1 = 72 + 13 = 85.

So let me compute the ratio:

[13 * (41/13) * (5185 / 2501)] = 41 * (5185 / 2501).

So need to check if 5185 / 2501 = 85 / something? Wait, 2501 * 2 = 5002, 2501 * 2.07... Let me check if 5185 = 2501 * something.

Alternatively, maybe 5185 = 8*648 + 1? Wait, 8*648 = 5184, so 5185 = 8*648 + 1.

But 2501 is 50^2 + 1. Not sure. Alternatively, maybe there's a relation between 5185 and 2501. Let me check if 5185 = 2501 * 2 + 183, as before.

Alternatively, maybe I made a mistake in the calculation. Let me think again.

Alternatively, since according to our formula, the answer should be 8n² + 4n + 1 = 8*9 + 12 + 1 = 85. Therefore, the ratio should be 85.

So let me compute 41 * (5185 / 2501). Let me compute 5185 divided by 2501. Let me note that 2501 * 2 = 5002, so 5185 - 5002 = 183. Therefore, 5185 / 2501 = 2 + 183/2501.

But 183 is 3*61, and 2501 is 50^2 + 1. Wait, 50^2 is 2500, so 2501 is 50^2 + 1. Not sure.

Alternatively, maybe there's a mistake in my previous steps? Let me check again.

Wait, if the formula is correct, then the ratio should be 85. Let me check if 41 * (5185 / 2501) equals 85.

Let me compute 41 * 5185 = ?

Well, 40 * 5185 = 207,400

Plus 1 * 5185 = 5,185

Total: 207,400 + 5,185 = 212,585

Now, divide by 2501: 212,585 / 2501. Let me check how many times 2501 goes into 212,585.

Well, 2501 * 80 = 200,080

Subtract: 212,585 - 200,080 = 12,505

Now, 2501 * 5 = 12,505.

Therefore, total is 80 + 5 = 85.

Yes! Therefore, 212,585 / 2501 = 85. Therefore, the ratio is indeed 85. Which matches the formula.

Therefore, the formula seems to hold.

Therefore, the simplified expression is 8n² + 4n + 1.

But let me check if this can be simplified further or written in another form.

Alternatively, maybe factor it? Let me see:

8n² + 4n + 1. Let me check discriminant: 16 - 32 = -16. So it doesn't factor over real numbers. So probably this is the simplest form.

Alternatively, maybe write it as (something)^2 + something? Not sure.

Alternatively, maybe notice that 8n² + 4n + 1 = (2n + 1)^2 + 2n^2? Let me check:

(2n + 1)^2 = 4n² + 4n + 1. Then adding 2n² gives 6n² + 4n + 1, which is not.

Alternatively, maybe 8n² + 4n + 1 = 4*(2n² + n) + 1. Not sure.

Alternatively, maybe write as 8n² + 4n + 1 = 8n² + 4n + 1.

Alternatively, maybe express in terms of (2n)^2 + something?

But since the problem says "simplify", and 8n² + 4n + 1 is a quadratic, and it's already simplified, so probably that is the answer.

But let me check again the steps to make sure I didn't make a mistake.

We started with the original expression, factored each term using Sophie Germain identity, noticed that the denominator terms had factors that canceled with numerator terms, leading to telescoping product, which resulted in A_n / A_0, which is 8n² + 4n + 1.

Yes, and verified with n=1, n=2, n=3. So seems correct.

Therefore, the simplified expression is 8n² + 4n + 1.

But let me check if there's another way to write this. For example, maybe in terms of (something else).

Alternatively, note that 8n² + 4n + 1 = 4*(2n² + n) + 1. Not sure.

Alternatively, maybe write as (2n + 1)^2 + 2n²? Wait:

(2n + 1)^2 = 4n² + 4n + 1. Then adding 2n² gives 6n² + 4n + 1. Not helpful.

Alternatively, maybe think of 8n² + 4n + 1 as (something else).

Alternatively, maybe write as 8n² + 4n + 1 = (something)^2 + ... but not sure.

Alternatively, maybe check if it's equal to (something related to the original terms).

But since the problem says "simplify", and 8n² + 4n + 1 is a simplified polynomial, I think that is the answer.

Therefore, the final answer is 8n² + 4n + 1, which can be written as \boxed{8n^2 + 4n + 1}.

**Final Answer**
\boxed{8n^2 + 4n + 1}

To simplify the expression

$$
\frac{(2^4 + \frac{1}{4})(4^4 + \frac{1}{4}) \dotsm [(2n)^4 + \frac{1}{4}]}{(1^4 + \frac{1}{4})(3^4 + \frac{1}{4}) \dotsm [(2n - 1)^4 + \frac{1}{4}]}
$$

we begin by recognizing a pattern in the terms. Each term in both the numerator and the denominator is of the form $ x^4 + \frac{1}{4} $, where $ x $ is an even or odd integer, respectively.

---

### Step 1: Factor the General Term

We use the **Sophie Germain identity**:

$$
x^4 + 4y^4 = (x^2 + 2xy + 2y^2)(x^2 - 2xy + 2y^2)
$$

Set $ y = \frac{1}{2} $, so that $ 4y^4 = \frac{1}{4} $. Then:

$$
x^4 + \frac{1}{4} = (x^2 + x + \frac{1}{2})(x^2 - x + \frac{1}{2})
$$

This allows us to factor each term in the numerator and denominator.

---

### Step 2: Apply the Factorization

Let’s denote:

- Numerator terms: $ (2k)^4 + \frac{1}{4} = (4k^2 + 2k + \frac{1}{2})(4k^2 - 2k + \frac{1}{2}) $
- Denominator terms: $ (2k - 1)^4 + \frac{1}{4} = (4k^2 - 2k + \frac{1}{2})(4k^2 - 6k + \frac{5}{2}) $

So the full expression becomes:

$$
\frac{\prod_{k=1}^n (4k^2 + 2k + \frac{1}{2})(4k^2 - 2k + \frac{1}{2})}{\prod_{k=1}^n (4k^2 - 2k + \frac{1}{2})(4k^2 - 6k + \frac{5}{2})}
$$

Notice that the second factor in the numerator, $ 4k^2 - 2k + \frac{1}{2} $, is the same as the first factor in the denominator. This allows for **cancellation** of the middle terms across the numerator and denominator.

After cancellation, the expression reduces to:

$$
\frac{\prod_{k=1}^n (4k^2 + 2k + \frac{1}{2})}{\prod_{k=1}^n (4k^2 - 6k + \frac{5}{2})}
$$

---

### Step 3: Express in Terms of a Single Polynomial

Let’s define:

- $ A_k = 4k^2 + 2k + \frac{1}{2} $
- $ D_k = 4k^2 - 6k + \frac{5}{2} $

We observe that:

$$
D_k = A_{k-1}
$$

This is verified by substituting $ k-1 $ into $ A_k $:

$$
A_{k-1} = 4(k-1)^2 + 2(k-1) + \frac{1}{2} = 4k^2 - 8k + 4 + 2k - 2 + \frac{1}{2} = 4k^2 - 6k + \frac{5}{2}
$$

So the denominator becomes:

$$
\prod_{k=1}^n D_k = \prod_{k=1}^n A_{k-1} = \prod_{m=0}^{n-1} A_m
$$

Thus, the entire expression becomes:

$$
\frac{\prod_{k=1}^n A_k}{\prod_{m=0}^{n-1} A_m} = \frac{A_n}{A_0}
$$

---

### Step 4: Evaluate the Final Expression

Recall:

$$
A_k = 4k^2 + 2k + \frac{1}{2}
$$

So:

- $ A_n = 4n^2 + 2n + \frac{1}{2} $
- $ A_0 = 4(0)^2 + 2(0) + \frac{1}{2} = \frac{1}{2} $

Therefore:

$$
\frac{A_n}{A_0} = \frac{4n^2 + 2n + \frac{1}{2}}{\frac{1}{2}} = 8n^2 + 4n + 1
$$

---

### Final Answer

$$
\boxed{8n^2 + 4n + 1}
$$

Solution:
Let
\[f(m) = m^4 + \frac{1}{4} = \frac{4m^4 + 1}{4}.\]We can factor this with a little give and take:
\begin{align*}
f(m) &= \frac{4m^4 + 1}{4} \\
&= \frac{4m^4 + 4m^2 + 1 - 4m^2}{4} \\
&= \frac{(2m^2 + 1)^2 - (2m)^2}{4} \\
&= \frac{(2m^2 + 2m + 1)(2m^2 - 2m + 1)}{4}.
\end{align*}Now, let $g(m) = 2m^2 + 2m + 1.$  Then
\[g(m - 1) = 2(m - 1)^2 + 2(m - 1) + 1 = 2m^2 - 2m + 1.\]Hence,
\[f(m) = \frac{g(m) g(m - 1)}{4}.\]Therefore,
\begin{align*}
\frac{(2^4 + \frac{1}{4})(4^4 + \frac{1}{4}) \dotsm [(2n)^4 + \frac{1}{4}]}{(1^4 + \frac{1}{4})(3^4 + \frac{1}{4}) \dotsm [(2n - 1)^4 + \frac{1}{4}]} &= \frac{f(2) f(4) \dotsm f(2n)}{f(1) f(3) \dotsm f(2n - 1)} \\
&= \frac{\frac{g(2) g(1)}{4} \cdot \frac{g(4) g(3)}{4} \dotsm \frac{g(2n) g(2n - 1)}{4}}{\frac{g(1) g(0)}{4} \cdot \frac{g(3) g(2)}{4} \dotsm \frac{g(2n - 1) g(2n - 2)}{4}} \\
&= \frac{g(2n)}{g(0)} \\
&= 2(2n)^2 + 2(2n) + 1 \\
&= \boxed{8n^2 + 4n + 1}.
\end{align*}


Explanation:
Okay, so I need to find the probability that the reciprocal of a randomly selected positive odd integer less than 2010 gives a terminating decimal. Then express that probability as a reduced fraction a/b and find a + b. Alright, let me start by recalling what makes a decimal terminating.

From what I remember, a decimal expansion of a fraction terminates if and only if the denominator, after simplifying the fraction to its lowest terms, has no prime factors other than 2 and 5. So, for example, 1/2 = 0.5, which terminates, and 1/3 = 0.333..., which doesn't. So, if I have a fraction 1/n, then 1/n will have a terminating decimal if and only if n is of the form 2^k * 5^m where k and m are non-negative integers. But wait, the question is about reciprocals of odd integers. So since the integers we are considering are odd, they are not divisible by 2. Therefore, for 1/n to have a terminating decimal, n must be of the form 5^m, right? Because the denominator can't have any prime factors other than 2 and 5, but since n is odd, it can't have 2 as a factor. Therefore, n must be a power of 5. Is that correct?

Let me check. Suppose n is an odd integer. Then, if n is a power of 5, say 5^m, then 1/n will terminate. If n has any other prime factors besides 5, then 1/n will not terminate. For example, 1/15 = 0.0666..., which doesn't terminate. So yes, that seems right. Therefore, the problem reduces to finding how many positive odd integers less than 2010 are powers of 5, and then dividing that by the total number of positive odd integers less than 2010. Then, simplify that fraction and find a + b.

Okay, so first step: find the total number of positive odd integers less than 2010. Since we're dealing with positive integers less than 2010, that's from 1 to 2009 inclusive. The number of odd numbers in this range is equal to the ceiling of 2009/2. Wait, but since 2009 is odd, the number of odd numbers is (2009 + 1)/2 = 2010/2 = 1005. Wait, let me check that again. The number of odd numbers between 1 and N inclusive is floor((N + 1)/2). If N is odd, then (N + 1)/2 is an integer. For example, N = 5: numbers 1, 3, 5: 3 numbers. (5 + 1)/2 = 3. So yes, for N = 2009, which is odd, the number of odd numbers is (2009 + 1)/2 = 2010/2 = 1005. So total number of possible integers is 1005. Got that.

Now, the next part is to find how many of these odd integers are powers of 5. So, we need to find all integers n such that n = 5^k, where n < 2010 and n is odd. Since 5^k is always odd, we don't have to worry about that. So, we need to find all exponents k such that 5^k < 2010.

Let me compute the powers of 5:

5^1 = 5

5^2 = 25

5^3 = 125

5^4 = 625

5^5 = 3125

Wait, 5^5 is 3125, which is greater than 2010. Therefore, the exponents k can be 1, 2, 3, 4. Therefore, there are 4 such numbers: 5, 25, 125, 625. Let me check:

5^1 = 5 < 2010

5^2 = 25 < 2010

5^3 = 125 < 2010

5^4 = 625 < 2010

5^5 = 3125 > 2010. So yes, only up to k=4. Therefore, there are 4 numbers. Therefore, the probability is 4 / 1005. Wait, but we need to check if 4 and 1005 are coprime. Let me check.

First, factorize 1005. 1005 divided by 5 is 201. Then 201 is 3*67. Therefore, 1005 = 5 * 3 * 67. The numerator is 4. The prime factors of 4 are 2^2. Since 4 and 1005 share no common prime factors, they are coprime. Therefore, the fraction is already reduced, so a = 4, b = 1005, so a + b = 4 + 1005 = 1009. Wait, but hold on. Is that correct? Let me check again.

Wait, but maybe I made a mistake here. Let me think again. The question says "positive odd integer less than 2010". So numbers from 1 to 2009 inclusive. But 5^k must be less than 2010. So 5^4 is 625, 5^5 is 3125. So 5^4 is 625, which is less than 2010. So that's four numbers: 5, 25, 125, 625. So that's four numbers. Therefore, the probability is 4 / 1005. But let me check if there are more numbers. Wait, is there another power of 5? For example, 5^0 is 1. Is 1 considered? Wait, the problem says "positive odd integer less than 2010". So 1 is included. But 1 is 5^0. However, does 1 count? Let me check.

If n = 1, then 1/1 = 1.0, which is a terminating decimal. So yes, 1 is a valid number. Therefore, 5^0 = 1 is also a power of 5. Therefore, maybe I missed that. So, the exponents k can be 0, 1, 2, 3, 4. Let me check 5^0 = 1, which is less than 2010. So that's another one. Therefore, there are 5 numbers: 1, 5, 25, 125, 625. Therefore, the count is 5 instead of 4. Therefore, the probability is 5 / 1005. Then, simplifying that fraction. Let me check.

So, 5 / 1005. Divide numerator and denominator by 5: 1 / 201. Therefore, a = 1, b = 201. Then a + b = 202. Wait, but why did I not consider 1 before? Because the problem says "positive odd integer less than 2010". So 1 is included. But when I thought about the reciprocal of 1, which is 1.0, which is terminating. Therefore, 1 should be included. Therefore, my initial count was wrong. So, there are 5 numbers: 1, 5, 25, 125, 625. Therefore, the probability is 5 / 1005. Then simplifying, divide numerator and denominator by 5: 1 / 201. Therefore, a = 1, b = 201. Then a + b = 202. But let me check again if there are more numbers. Let me check 5^5 is 3125, which is more than 2010, so no. So 5^0 is 1, 5^1 is 5, 5^2 is 25, 5^3 is 125, 5^4 is 625. So that's five numbers. Therefore, the count is 5. Therefore, probability is 5 / 1005 = 1 / 201. Therefore, a + b = 1 + 201 = 202. But let me check again if 1 is considered. The problem says "positive odd integer less than 2010". So 1 is a positive odd integer less than 2010. Therefore, yes, it should be included. Therefore, the answer is 202? Wait, but let me check again.

Wait, but maybe there's a mistake here. Let me think again. The original reasoning was that for 1/n to have a terminating decimal, n must be of the form 2^k * 5^m. However, since n is odd, it cannot have factors of 2, so m must be such that n is 5^m. However, 1 is 5^0, so that's allowed. Therefore, n can be 5^0, 5^1, 5^2, etc. So, in that case, the numbers are 1, 5, 25, 125, 625. So five numbers. Therefore, the count is 5. Therefore, probability is 5 / 1005. Simplify that fraction. Let me check if 5 divides into 1005. 1005 divided by 5 is 201. So yes, 5 / 1005 = 1 / 201. Therefore, a = 1, b = 201. Therefore, a + b = 202. But let me check if there are other numbers. For example, numbers that are products of 5s and 1s? Wait, but since n is odd, and we need n to be of the form 5^m. Because if n had any other prime factors, then 1/n would not terminate. Therefore, only numbers that are powers of 5. Therefore, only 1, 5, 25, 125, 625. Therefore, five numbers. Therefore, the answer is 202. But let me check again if 1 is considered. Let me check the problem statement again: "the reciprocal of a randomly selected positive odd integer less than 2010 gives a terminating decimal". So reciprocal of 1 is 1.0, which is terminating. So yes, 1 is included. Therefore, the count is 5. Therefore, probability is 5 / 1005 = 1 / 201. Therefore, answer is 1 + 201 = 202. But wait, let me check if there are more numbers. For example, numbers like 5^0 = 1, 5^1 = 5, 5^2 = 25, 5^3 = 125, 5^4 = 625, 5^5 = 3125. 3125 is more than 2010, so no. Therefore, only five numbers. Therefore, 5 numbers. Therefore, 5 / 1005. Simplify. 5 divides into 1005, as 1005 / 5 = 201. So 5 / 1005 = 1 / 201. Therefore, a = 1, b = 201. Therefore, a + b = 202. Therefore, the answer is 202. But let me check again if there's a mistake here. Maybe I need to check if there are other numbers which are not powers of 5 but still have only 5 as prime factors? Wait, but if n is an odd integer, then n can't have 2 as a factor. So if n is to have only 5 as prime factors, then n must be 5^k. Therefore, there are no other numbers. For example, 25 is 5^2, 125 is 5^3, etc. So, no other numbers. Therefore, only 1, 5, 25, 125, 625. Therefore, five numbers. Therefore, the answer is 202. Wait, but let me check with another approach. Let me think again.

Another way: The probability is equal to the number of odd integers less than 2010 which are of the form 5^k divided by the total number of odd integers less than 2010. So, total number of odd integers less than 2010 is 1005 as before. Number of such n is 5. Therefore, 5 / 1005. Simplify. 5 divides into 1005, so divide numerator and denominator by 5: 1 / 201. Therefore, 1 + 201 = 202. Therefore, answer is 202. But let me check if 1 is considered. Suppose someone argues that 1 is not considered because it's trivial? But the problem says "positive odd integer", so 1 is included. Therefore, I think that is correct. However, let me check with an example. Let me take a smaller number instead of 2010. Let's say, instead of less than 2010, let's say less than 10. Then, the odd numbers less than 10 are 1, 3, 5, 7, 9. The reciprocals: 1/1 = 1.0, terminating; 1/3 = 0.333..., non-terminating; 1/5 = 0.2, terminating; 1/7 ≈ 0.1428..., non-terminating; 1/9 ≈ 0.111..., non-terminating. Therefore, there are two numbers: 1 and 5. Therefore, probability is 2 / 5. According to our formula, the number of powers of 5 less than 10: 5^0 = 1, 5^1 = 5, 5^2 = 25 which is more than 10. Therefore, two numbers. Total odd numbers less than 10 is 5. Therefore, 2 / 5. Which is correct. Therefore, in this case, the answer would be 2 / 5, which reduces to 2 / 5, so a + b = 7. So, in this case, 1 is included. Therefore, seems like 1 is included. Therefore, in our original problem, 1 is included. Therefore, answer is 202. Therefore, I think that is correct. Therefore, the answer is 202. Wait, but let me check another example. Let me take numbers less than 25. Then, odd numbers less than 25: 1, 3, 5, 7, 9, 11, 13, 15, 17, 19, 21, 23. Total of 12 numbers. The numbers which are powers of 5: 1, 5, 25. But 25 is not less than 25, so only 1 and 5. Therefore, 2 numbers. Therefore, probability is 2 / 12 = 1 / 6. According to our formula, number of powers of 5 less than 25: 5^0 = 1, 5^1 = 5, 5^2 = 25 which is not less than 25. Therefore, two numbers. Therefore, 2 / 12 = 1 / 6. Which is correct. Therefore, in this case, 1 is included. Therefore, seems like 1 is included. Therefore, in our original problem, 1 is included. Therefore, answer is 202. Therefore, I think that is correct. Therefore, the answer is 202. However, let me check once again if there are other numbers. For example, numbers like 5^k multiplied by 1? But 1 is already included. Wait, no. If n is 5^k, then n is already a power of 5. So, there are no other numbers. Therefore, only 1, 5, 25, 125, 625. Therefore, five numbers. Therefore, 5 / 1005 = 1 / 201. Therefore, answer is 202. Therefore, I think that is correct. Therefore, the answer is 202. Wait, but let me check if 1 is considered. Suppose someone says that 1 is not considered because it's not a "positive odd integer greater than 1"? But the problem says "positive odd integer less than 2010", so 1 is included. Therefore, I think that is correct. Therefore, I think the answer is 202. Therefore, I think the answer is \boxed{202}.

**Final Answer**
\boxed{202}

To solve the problem, we are asked to find the probability that the reciprocal of a randomly selected **positive odd integer less than 2010** gives a **terminating decimal**, and then express this probability as a reduced fraction $ \frac{a}{b} $, where $ a $ and $ b $ are relatively prime positive integers. Finally, we are to compute $ a + b $.

---

### Step 1: Understanding Terminating Decimals

A fraction $ \frac{1}{n} $ has a **terminating decimal** if and only if the **denominator**, when the fraction is in lowest terms, has **no prime factors other than 2 and 5**. That is, $ n $ must be of the form $ 2^k \cdot 5^m $, where $ k, m \geq 0 $.

However, in this problem, we are only considering **odd integers**, so $ n $ cannot be divisible by 2. Therefore, the only allowed prime factor is **5**. Thus, for $ \frac{1}{n} $ to have a terminating decimal, $ n $ must be a **power of 5**.

---

### Step 2: Counting the Total Number of Positive Odd Integers Less Than 2010

We are selecting from all **positive odd integers less than 2010**. These are the numbers:

$$
1, 3, 5, 7, \ldots, 2009
$$

This is an arithmetic sequence with first term 1, last term 2009, and common difference 2. The number of terms is:

$$
\frac{2009 - 1}{2} + 1 = \frac{2008}{2} + 1 = 1004 + 1 = 1005
$$

So, there are **1005** such numbers.

---

### Step 3: Counting the Number of Powers of 5 Less Than 2010

We now count how many **odd integers less than 2010** are powers of 5. Since all powers of 5 are odd, we only need to find the number of powers of 5 less than 2010.

Compute successive powers of 5:

- $ 5^0 = 1 $
- $ 5^1 = 5 $
- $ 5^2 = 25 $
- $ 5^3 = 125 $
- $ 5^4 = 625 $
- $ 5^5 = 3125 $ (which is **greater than 2010**)

So, the valid powers of 5 less than 2010 are:

$$
1, 5, 25, 125, 625
$$

That is, **5** such numbers.

---

### Step 4: Calculating the Probability

The probability that a randomly selected odd integer less than 2010 has a reciprocal that is a terminating decimal is:

$$
\frac{5}{1005}
$$

Now, simplify the fraction:

$$
\frac{5}{1005} = \frac{1}{201}
$$

So, $ a = 1 $, $ b = 201 $, and $ a + b = 1 + 201 = \boxed{202} $

---

### Final Answer

$$
\boxed{202}
$$

[Assistant]
Explanation:
Okay, so I need to find the probability that the reciprocal of a randomly selected positive odd integer less than 2010 gives a terminating decimal. Then express that probability as a reduced fraction a/b and find a + b. Alright, let me start by recalling what makes a decimal terminating.

From what I remember, a decimal expansion of a fraction terminates if and only if the denominator, after simplifying the fraction to its lowest terms, has no prime factors other than 2 and 5. So, for example, 1/2 = 0.5, which terminates, and 1/3 = 0.333..., which doesn't. So, if I have a fraction 1/n, then 1/n will have a terminating decimal if and only if n is of the form 2^k * 5^m where k and m are non-negative integers. But wait, the question is about reciprocals of odd integers. So since the integers we are considering are odd, they are not divisible by 2. Therefore, for 1/n to have a terminating decimal, n must be of the form 5^m, right? Because the denominator can't have any prime factors other than 2 and 5, but since n is odd, it can't have 2 as a factor. Therefore, n must be a power of 5. Is that correct?

Let me check. Suppose n is an odd integer. Then, if n is a power of 5, say 5^m, then 1/n will terminate. If n has any other prime factors besides 5, then 1/n will not terminate. For example, 1/15 = 0.0666..., which doesn't terminate. So yes, that seems right. Therefore, the problem reduces to finding how many positive odd integers less than 2010 are powers of 5, and then dividing that by the total number of positive odd integers less than 2010. Then, simplify that fraction and find a + b.

Okay, so first step: find the total number of positive odd integers less than 2010. Since we're dealing with positive integers less than 2010, that's from 1 to 2009 inclusive. The number of odd numbers in this range is equal to the ceiling of 2009/2. Wait, but since 2009 is odd, the number of odd numbers is (2009 + 1)/2 = 2010/2 = 1005. Wait, let me check that again. The number of odd numbers between 1 and N inclusive is floor((N + 1)/2). If N is odd, then (N + 1)/2 is an integer. For example, N = 5: numbers 1, 3, 5: 3 numbers. (5 + 1)/2 = 3. So yes, for N = 2009, which is odd, the number of odd numbers is (2009 + 1)/2 = 2010/2 = 1005. So total number of possible integers is 1005. Got that.

Now, the next part is to find how many of these odd integers are powers of 5. So, we need to find all integers n such that n = 5^k, where n < 2010 and n is odd. Since 5^k is always odd, we don't have to worry about that. So, we need to find all exponents k such that 5^k < 2010.

Let me compute the powers of 5:

5^1 = 5

5^2 = 25

5^3 = 125

5^4 = 625

5^5 = 3125

Wait, 5^5 is 3125, which is greater than 2010. Therefore, the exponents k can be 1, 2, 3, 4. Therefore, there are 4 such numbers: 5, 25, 125, 625. Let me check:

5^1 = 5 < 2010

5^2 = 25 < 2010

5^3 = 125 < 2010

5^4 = 625 < 2010

5^5 = 3125 > 2010. So yes, only up to k=4. Therefore, there are 4 numbers. Therefore, the probability is 4 / 1005. Wait, but we need to check if 4 and 1005 are coprime. Let me check.

First, factorize 1005. 1005 divided by 5 is 201. Then 201 is 3*67. Therefore, 1005 = 5 * 3 * 67. The numerator is 4. The prime factors of 4 are 2^2. Since 4 and 1005 share no common prime factors, they are coprime. Therefore, the fraction is already reduced, so a = 4, b = 1005, so a + b = 4 + 1005 = 1009. Wait, but hold on. Is that correct? Let me check again.

Wait, but maybe I made a mistake here. Let me think again. The question says "positive odd integer less than 2010". So numbers from 1 to 2009 inclusive. But 5^k must be less than 2010. So 5^4 is 625, 5^5 is 3125. So 5^4 is 625, which is less than 2010. So that's four numbers: 5, 25, 125, 625. So that's four numbers. Therefore, the probability is 4 / 1005. But let me check if there are more numbers. Wait, is there another power of 5? For example, 5^0 is 1. Is 1 considered? Wait, the problem says "positive odd integer less than 2010". So 1 is included. But 1 is 5^0. However, does 1 count? Let me check.

If n = 1, then 1/1 = 1.0, which is a terminating decimal. So yes, 1 is a valid number. Therefore, 5^0 = 1 is also a power of 5. Therefore, maybe I missed that. So, the exponents k can be 0, 1, 2, 3, 4. Let me check 5^0 = 1, which is less than 2010. So that's another one. Therefore, there are 5 numbers: 1, 5, 25, 125, 625. Therefore, the count is 5 instead of 4. Therefore, the probability is 5 / 1005. Then, simplifying that fraction. Let me check.

So, 5 / 1005. Divide numerator and denominator by 5: 1 / 201. Therefore, a = 1, b = 201. Then a + b = 202. Wait, but why did I not consider 1 before? Because the problem says "positive odd integer less than 2010". So 1 is included. But when I thought about the reciprocal of 1, which is 1.0, which is terminating. Therefore, 1 should be included. Therefore, my initial count was wrong. So, there are 5 numbers: 1, 5, 25, 125, 625. Therefore, the probability is 5 / 1005. Then simplifying, divide numerator and denominator by 5: 1 / 201. Therefore, a = 1, b = 201. Then a + b = 202. But let me check again if there are more numbers. Let me check 5^5 is 3125, which is more than 2010, so no. So 5^0 is 1, 5^1 is 5, 5^2 is 25, 5^3 is 125, 5^4 is 625. So that's five numbers. Therefore, the count is 5. Therefore, probability is 5 / 1005 = 1 / 201. Therefore, a + b = 1 + 201 = 202. But let me check again if 1 is considered. The problem says "positive odd integer less than 2010". So 1 is a positive odd integer less than 2010. Therefore, yes, it should be included. Therefore, the answer is 202? Wait, but let me check again.

Wait, but maybe there's a mistake here. Let me think again. The original reasoning was that for 1/n to have a terminating decimal, n must be of the form 2^k * 5^m. However, since n is odd, it cannot have factors of 2, so m must be such that n is 5^m. However, 1 is 5^0, so that's allowed. Therefore, n can be 5^0, 5^1, 5^2, etc. So, in that case, the numbers are 1, 5, 25, 125, 625. So five numbers. Therefore, the count is 5. Therefore, probability is 5 / 1005. Simplify that fraction. Let me check if 5 divides into 1005. 1005 divided by 5 is 201. So yes, 5 / 1005 = 1 / 201. Therefore, a = 1, b = 201. Therefore, a + b = 202. But let me check if there are other numbers. For example, numbers that are products of 5s and 1s? Wait, but since n is odd, and we need n to be of the form 5^m. Because if n had any other prime factors, then 1/n would not terminate. Therefore, only numbers that are powers of 5. Therefore, only 1, 5, 25, 125, 625. Therefore, five numbers. Therefore, the answer is 202. But let me check again if 1 is considered. Let me check the problem statement again: "the reciprocal of a randomly selected positive odd integer less than 2010 gives a terminating decimal". So reciprocal of 1 is 1.0, which is terminating. So yes, 1 is included. Therefore, the count is 5. Therefore, probability is 5 / 1005 = 1 / 201. Therefore, answer is 1 + 201 = 202. But wait, let me check if there are more numbers. For example, numbers like 5^0 = 1, 5^1 = 5, 5^2 = 25, 5^3 = 125, 5^4 = 625, 5^5 = 3125. 3125 is more than 2010, so no. Therefore, only five numbers. Therefore, 5 numbers. Therefore, 5 / 1005. Simplify. 5 divides into 1005, as 1005 / 5 = 201. So 5 / 1005 = 1 / 201. Therefore, a = 1, b = 201. Therefore, a + b = 202. Therefore, the answer is 202. But let me check again if there's a mistake here. Maybe I need to check if there are other numbers which are not powers of 5 but still have only 5 as prime factors? Wait, but if n is an odd integer, then n can't have 2 as a factor. So if n is to have only 5 as prime factors, then n must be 5^k. Therefore, there are no other numbers. For example, 25 is 5^2, 125 is 5^3, etc. So, no other numbers. Therefore, only 1, 5, 25, 125, 625. Therefore, five numbers. Therefore, the answer is 202. Wait, but let me check with another approach. Let me think again.

Another way: The probability is equal to the number of odd integers less than 2010 which are of the form 5^k divided by the total number of odd integers less than 2010. So, total number of odd integers less than 2010 is 1005 as before. Number of such n is 5. Therefore, 5 / 1005. Simplify. 5 divides into 1005, so divide numerator and denominator by 5: 1 / 201. Therefore, 1 + 201 = 202. Therefore, answer is 202. But let me check if 1 is considered. Suppose someone argues that 1 is not considered because it's trivial? But the problem says "positive odd integer", so 1 is included. Therefore, I think that is correct. However, let me check with an example. Let me take a smaller number instead of 2010. Let's say, instead of less than 2010, let's say less than 10. Then, the odd numbers less than 10 are 1, 3, 5, 7, 9. The reciprocals: 1/1 = 1.0, terminating; 1/3 = 0.333..., non-terminating; 1/5 = 0.2, terminating; 1/7 ≈ 0.1428..., non-terminating; 1/9 ≈ 0.111..., non-terminating. Therefore, there are two numbers: 1 and 5. Therefore, probability is 2 / 5. According to our formula, the number of powers of 5 less than 10: 5^0 = 1, 5^1 = 5, 5^2 = 25 which is more than 10. Therefore, two numbers. Total odd numbers less than 10 is 5. Therefore, 2 / 5. Which is correct. Therefore, in this case, the answer would be 2 / 5, which reduces to 2 / 5, so a + b = 7. So, in this case, 1 is included. Therefore, seems like 1 is included. Therefore, in our original problem, 1 is included. Therefore, answer is 202. Therefore, I think that is correct. Therefore, the answer is 202. Wait, but let me check another example. Let me take numbers less than 25. Then, odd numbers less than 25: 1, 3, 5, 7, 9, 11, 13, 15, 17, 19, 21, 23. Total of 12 numbers. The numbers which are powers of 5: 1, 5, 25. But 25 is not less than 25, so only 1 and 5. Therefore, 2 numbers. Therefore, probability is 2 / 12 = 1 / 6. According to our formula, number of powers of 5 less than 25: 5^0 = 1, 5^1 = 5, 5^2 = 25 which is not less than 25. Therefore, two numbers. Therefore, 2 / 12 = 1 / 6. Which is correct. Therefore, in this case, 1 is included. Therefore, seems like 1 is included. Therefore, in our original problem, 1 is included. Therefore, answer is 202. Therefore, I think that is correct. Therefore, the answer is 202. However, let me check once again if there are other numbers. For example, numbers like 5^k multiplied by 1? But 1 is already included. Wait, no. If n is 5^k, then n is already a power of 5. So, there are no other numbers. Therefore, only 1, 5, 25, 125, 625. Therefore, five numbers. Therefore, 5 / 1005 = 1 / 201. Therefore, answer is 202. Therefore, I think that is correct. Therefore, the answer is 202. Wait, but let me check if 1 is considered. Suppose someone says that 1 is not considered because it's not a "positive odd integer greater than 1"? But the problem says "positive odd integer less than 2010", so 1 is included. Therefore, I think that is correct. Therefore, I think the answer is 202. Therefore, I think the answer is \boxed{202}.

**Final Answer**
\boxed{202}

To solve the problem, we are asked to find the probability that the reciprocal of a randomly selected **positive odd integer less than 2010** gives a **terminating decimal**, and then express this probability as a reduced fraction $ \frac{a}{b} $, where $ a $ and $ b $ are relatively prime positive integers. Finally, we are to compute $ a + b $.

---

### Step 1: Understanding Terminating Decimals

A fraction $ \frac{1}{n} $ has a **terminating decimal** if and only if the **denominator**, when the fraction is in lowest terms, has **no prime factors other than 2 and 5**. That is, $ n $ must be of the form $ 2^k \cdot 5^m $, where $ k, m \geq 0 $.

However, in this problem, we are only considering **odd integers**, so $ n $ cannot be divisible by 2. Therefore, the only allowed prime factor is **5**. Thus, for $ \frac{1}{n} $ to have a terminating decimal, $ n $ must be a **power of 5**.

---

### Step 2: Counting the Total Number of Positive Odd Integers Less Than 2010

We are selecting from all **positive odd integers less than 2010**. These are the numbers:

$$
1, 3, 5, 7, \ldots, 2009
$$

This is an arithmetic sequence with first term 1, last term 2009, and common difference 2. The number of terms is:

$$
\frac{2009 - 1}{2} + 1 = \frac{2008}{2} + 1 = 1004 + 1 = 1005
$$

So, there are **1005** such numbers.

---

### Step 3: Counting the Number of Powers of 5 Less Than 2010

We now count how many **odd integers less than 2010** are powers of 5. Since all powers of 5 are odd, we only need to find the number of powers of 5 less than 2010.

Compute successive powers of 5:

- $ 5^0 = 1 $
- $ 5^1 = 5 $
- $ 5^2 = 25 $
- $ 5^3 = 125 $
- $ 5^4 = 625 $
- $ 5^5 = 3125 $ (which is **greater than 2010**)

So, the valid powers of 5 less than 2010 are:

$$
1, 5, 25, 125, 625
$$

That is, **5** such numbers.

---

### Step 4: Calculating the Probability

The probability that a randomly selected odd integer less than 2010 has a reciprocal that is a terminating decimal is:

$$
\frac{5}{1005}
$$

Now, simplify the fraction:

$$
\frac{5}{1005} = \frac{1}{201}
$$

So, $ a = 1 $, $ b = 201 $, and $ a + b = 1 + 201 = \boxed{202} $

---

### Final Answer

$$
\boxed{202}
$$

Solution:
There are 2009 positive integers less than 2010, of which 1005 are odd. If $\frac{1}{n}$ is equal to a terminating decimal, then $n$ can only be divisible by 2 and 5. However, since we have the added restriction that $n$ is odd, $n$ must be a power of 5. There are five powers of 5 less than 2010. \begin{align*}
5^0 &= 1 \\
5^1 &= 5 \\
5^2 &= 25 \\
5^3 &= 125 \\
5^4 &= 625
\end{align*} Note that $5^5 = 3125$. Since there are five odd integers that satisfy our desired condition, the desired probability is $\frac{5}{1005} = \frac{1}{201}$. This is in simplest terms, so our answer is $1+201 = \boxed{202}$.


Explanation:
Okay, so I need to find the original price of a shirt when the sale price is $14.40, and that sale price is 60% off the original price. Hmm, let me think.

First, let me make sure I understand what "60% off" means. If something is 60% off, that means the customer is paying 40% of the original price, right? Because 100% - 60% = 40%. So the sale price is 40% of the original price.

So, if I let the original price be some amount, let's call it "P", then 40% of P equals $14.40.

Mathematically, that would be 0.40 * P = 14.40.

So, to find P, I need to divide both sides of the equation by 0.40.

Let me write that down:

P = 14.40 / 0.40

Hmm, let me calculate that. Dividing by 0.40 is the same as multiplying by 2.5, right? Because 1 divided by 0.4 is 2.5. Let me check that. 0.4 times 2.5 is 1.0, yes. So 14.40 divided by 0.40 is 14.40 * 2.5.

Let me compute 14.40 * 2.5. Breaking it down, 14.40 * 2 is 28.80, and 14.40 * 0.5 is 7.20. Adding those together: 28.80 + 7.20 = 36.00.

So, the original price was $36.00? Let me verify that. If the original price is $36, then 60% off would be 0.60 * 36 = 21.60. Subtracting that discount from the original price: 36 - 21.60 = 14.40. Yep, that matches the sale price given.

Wait, but let me make sure I didn't mix up anything. Sometimes when people say "discount," they might mean different things, but in this case, it's straightforward. 60% off means the discount is 60%, so the sale price is original minus 60% of original, which is 40% of original.

Alternatively, another way to think about it is if the sale price is 40% of the original, then original price is sale price divided by 0.4. Which is exactly what I did.

So, 14.40 divided by 0.4. Let me do that division again to confirm.

Dividing 14.40 by 0.4. Let me convert them into fractions to make it clearer. 14.40 is the same as 1440 cents, and 0.4 is 40 cents. So 1440 divided by 40 is 36. So that's 36 dollars.

Alternatively, moving the decimal: 14.40 divided by 0.4. Multiply numerator and denominator by 10 to eliminate the decimal: 144.0 divided by 4. Which is 36. Yep, same result.

So, all methods point to the original price being $36.

Wait, but let me think again. If the sale price is 60% off, does that mean that the sale price is 60% of the original price? No, that would be a common mistake. If something is 60% off, you're taking away 60%, so the remaining is 40%. So, sale price is 40% of original.

But just to make sure, if someone says "60% off," does that mean the price is reduced by 60%, so the new price is original minus 60% of original? Yes. So, original price times (1 - 0.60) = 0.40 * original price.

So, that's correct. Therefore, solving for original price, we have original price = sale price / 0.40.

So, 14.40 / 0.40 = 36.

Therefore, the original price was $36.

I think that's solid. Let me check with another approach. Let's assume original price is $36.

Calculating 60% of $36: 0.60 * 36 = 21.60.

Subtracting that from original price: 36 - 21.60 = 14.40.

Which is exactly the sale price given.

Therefore, the original price must have been $36.

I don't see any mistakes in my reasoning. So, I think the answer is $36.

**Final Answer**
The original price of the shirt was \boxed{36} dollars.

To determine the original price of the shirt, we start by understanding the relationship between the sale price and the original price.

---

### Step 1: Understand the Discount

The problem states that the sale price is **60% off** the original price. This means the customer is paying **40%** of the original price. In mathematical terms:

$$
\text{Sale Price} = 40\% \times \text{Original Price}
$$

Or, using decimals:

$$
\text{Sale Price} = 0.40 \times \text{Original Price}
$$

---

### Step 2: Set Up the Equation

We are given that the sale price is **$14.40**, so we can write:

$$
0.40 \times P = 14.40
$$

Where $ P $ is the original price.

---

### Step 3: Solve for the Original Price

To isolate $ P $, divide both sides of the equation by 0.40:

$$
P = \frac{14.40}{0.40}
$$

$$
P = 36
$$

---

### Step 4: Verify the Result

Let’s confirm by calculating 60% off of $36:

$$
60\% \text{ of } 36 = 0.60 \times 36 = 21.60
$$

Subtracting the discount from the original price:

$$
36 - 21.60 = 14.40
$$

This matches the given sale price, confirming our calculation is correct.

---

### Final Answer

$$
\boxed{36}
$$

[Assistant]
Explanation:
Okay, so I need to find the original price of a shirt when the sale price is $14.40, and that sale price is 60% off the original price. Hmm, let me think.

First, let me make sure I understand what "60% off" means. If something is 60% off, that means the customer is paying 40% of the original price, right? Because 100% - 60% = 40%. So the sale price is 40% of the original price.

So, if I let the original price be some amount, let's call it "P", then 40% of P equals $14.40.

Mathematically, that would be 0.40 * P = 14.40.

So, to find P, I need to divide both sides of the equation by 0.40.

Let me write that down:

P = 14.40 / 0.40

Hmm, let me calculate that. Dividing by 0.40 is the same as multiplying by 2.5, right? Because 1 divided by 0.4 is 2.5. Let me check that. 0.4 times 2.5 is 1.0, yes. So 14.40 divided by 0.40 is 14.40 * 2.5.

Let me compute 14.40 * 2.5. Breaking it down, 14.40 * 2 is 28.80, and 14.40 * 0.5 is 7.20. Adding those together: 28.80 + 7.20 = 36.00.

So, the original price was $36.00? Let me verify that. If the original price is $36, then 60% off would be 0.60 * 36 = 21.60. Subtracting that discount from the original price: 36 - 21.60 = 14.40. Yep, that matches the sale price given.

Wait, but let me make sure I didn't mix up anything. Sometimes when people say "discount," they might mean different things, but in this case, it's straightforward. 60% off means the discount is 60%, so the sale price is original minus 60% of original, which is 40% of original.

Alternatively, another way to think about it is if the sale price is 40% of the original, then original price is sale price divided by 0.4. Which is exactly what I did.

So, 14.40 divided by 0.4. Let me do that division again to confirm.

Dividing 14.40 by 0.4. Let me convert them into fractions to make it clearer. 14.40 is the same as 1440 cents, and 0.4 is 40 cents. So 1440 divided by 40 is 36. So that's 36 dollars.

Alternatively, moving the decimal: 14.40 divided by 0.4. Multiply numerator and denominator by 10 to eliminate the decimal: 144.0 divided by 4. Which is 36. Yep, same result.

So, all methods point to the original price being $36.

Wait, but let me think again. If the sale price is 60% off, does that mean that the sale price is 60% of the original price? No, that would be a common mistake. If something is 60% off, you're taking away 60%, so the remaining is 40%. So, sale price is 40% of original.

But just to make sure, if someone says "60% off," does that mean the price is reduced by 60%, so the new price is original minus 60% of original? Yes. So, original price times (1 - 0.60) = 0.40 * original price.

So, that's correct. Therefore, solving for original price, we have original price = sale price / 0.40.

So, 14.40 / 0.40 = 36.

Therefore, the original price was $36.

I think that's solid. Let me check with another approach. Let's assume original price is $36.

Calculating 60% of $36: 0.60 * 36 = 21.60.

Subtracting that from original price: 36 - 21.60 = 14.40.

Which is exactly the sale price given.

Therefore, the original price must have been $36.

I don't see any mistakes in my reasoning. So, I think the answer is $36.

**Final Answer**
The original price of the shirt was \boxed{36} dollars.

To determine the original price of the shirt, we start by understanding the relationship between the sale price and the original price.

---

### Step 1: Understand the Discount

The problem states that the sale price is **60% off** the original price. This means the customer is paying **40%** of the original price. In mathematical terms:

$$
\text{Sale Price} = 40\% \times \text{Original Price}
$$

Or, using decimals:

$$
\text{Sale Price} = 0.40 \times \text{Original Price}
$$

---

### Step 2: Set Up the Equation

We are given that the sale price is **$14.40**, so we can write:

$$
0.40 \times P = 14.40
$$

Where $ P $ is the original price.

---

### Step 3: Solve for the Original Price

To isolate $ P $, divide both sides of the equation by 0.40:

$$
P = \frac{14.40}{0.40}
$$

$$
P = 36
$$

---

### Step 4: Verify the Result

Let’s confirm by calculating 60% off of $36:

$$
60\% \text{ of } 36 = 0.60 \times 36 = 21.60
$$

Subtracting the discount from the original price:

$$
36 - 21.60 = 14.40
$$

This matches the given sale price, confirming our calculation is correct.

---

### Final Answer

$$
\boxed{36}
$$

Solution:
If the shirt is $60\%$ off, it is currently $.4$ of the original price.  Thus the original price was

$$\frac{\$14.40}{.4}=\boxed{\$36}$$