Explanation:
Okay, so I need to find the smallest positive integer k such that when I multiply 240 by k, the result is a perfect cube. Hmm, let me think about how to approach this.

First, I remember that a perfect cube is a number that can be expressed as some integer raised to the power of three. For example, 8 is a perfect cube because it's 2³, and 27 is 3³. So, if 240 times k is a perfect cube, then all the prime factors in its prime factorization must have exponents that are multiples of three.

Alright, so maybe I should start by factoring 240 into its prime factors. Let me do that.

Breaking down 240:

240 divided by 2 is 120.
120 divided by 2 is 60.
60 divided by 2 is 30.
30 divided by 2 is 15.
15 divided by 3 is 5.
5 divided by 5 is 1.

So, the prime factors of 240 are 2⁴ × 3¹ × 5¹. Let me check that again.

Yes, 2⁴ is 16, 3¹ is 3, 5¹ is 5. Multiplying them together: 16 × 3 = 48, 48 × 5 = 240. Perfect.

So, 240 = 2⁴ × 3¹ × 5¹.

Now, if I multiply this by k, the product should be a perfect cube. Let me denote the prime factorization of k as something. Let me think.

Suppose k has prime factors 2^a × 3^b × 5^c × ... (other primes if necessary). Then, multiplying by 240 would give:

(2⁴ × 3¹ × 5¹) × (2^a × 3^b × 5^c × ...) = 2^(4+a) × 3^(1+b) × 5^(1+c) × ...

For this product to be a perfect cube, each exponent in the prime factorization must be a multiple of 3.

So, for each prime, the exponent after multiplying by k must be divisible by 3.

So, let's consider each prime factor:

For prime 2: exponent is 4 + a. This needs to be a multiple of 3.

Similarly, for prime 3: exponent is 1 + b. Needs to be multiple of 3.

For prime 5: exponent is 1 + c. Needs to be multiple of 3.

And for any other primes in k, their exponents must also be multiples of 3.

But since we want the smallest possible k, we should avoid introducing any new primes. Because adding new primes would require their exponents to be multiples of 3, which would make k larger. So, probably k only needs to have primes 2, 3, and 5.

So, let me focus on those.

Starting with prime 2: 4 + a ≡ 0 mod 3.

So, 4 + a must be divisible by 3. Let me solve for a.

What's the smallest non-negative integer a such that 4 + a is divisible by 3?

Well, 4 divided by 3 leaves a remainder of 1. So, to make it divisible by 3, we need to add 2, because 1 + 2 = 3. So, a = 2.

Wait, let me check: 4 + 2 = 6, which is 3×2. Yes, that works.

Alternatively, if a = 2, then exponent of 2 becomes 6, which is 3×2.

But is there a smaller a? Well, a has to be non-negative. If a = -1, that would be 3, but negative exponents aren't allowed because k is a positive integer. So, the smallest a is 2.

Okay, moving on to prime 3: exponent is 1 + b.

Similarly, 1 + b must be divisible by 3.

So, 1 + b ≡ 0 mod 3 ⇒ b ≡ -1 mod 3 ⇒ b ≡ 2 mod 3.

So, the smallest non-negative integer b is 2.

Same with prime 5: exponent is 1 + c. So, same as prime 3. So, c = 2.

Therefore, k would be 2² × 3² × 5².

Wait, let me check that.

So, if k is 2² × 3² × 5², then multiplying by 240:

Original exponents: 2⁴ × 3¹ × 5¹.

After multiplying by k: 2^(4+2) × 3^(1+2) × 5^(1+2) = 2^6 × 3^3 × 5^3.

Which is (2² × 3 × 5)^3.

Yes, because 2^6 is (2²)^3, 3^3 is (3)^3, 5^3 is (5)^3. So, altogether, it's (2² × 3 × 5)^3.

So, that's a perfect cube.

But wait, is this the smallest possible k? Let me check if there's a smaller k.

Wait, maybe I can have exponents that are higher multiples? For example, for prime 2, instead of adding 2 to get 6, could I add 5 to get 9? But that would make k larger. Similarly for others. So, the minimal exponents are 2 for each.

But let me think again.

Alternatively, maybe there's a way to have exponents that are multiples of 3 by subtracting? But since exponents can't be negative, we can't subtract.

So, for each prime, we need to find the minimal exponent to add to make the total exponent a multiple of 3.

So, for prime 2: original exponent is 4. The next multiple of 3 after 4 is 6. So, need to add 2.

For prime 3: original exponent is 1. Next multiple of 3 is 3. So, need to add 2.

Same for prime 5: original exponent is 1. Next multiple of 3 is 3. So, need to add 2.

Therefore, k is 2² × 3² × 5².

Calculating that: 2² is 4, 3² is 9, 5² is 25.

So, 4 × 9 = 36, 36 × 25 = 900.

So, k is 900?

Wait, but let me check if there's a smaller k.

Is there a possibility that some exponents can be adjusted in a way that k is smaller?

For example, maybe if we have some exponents that are higher multiples?

Wait, but if we take exponents higher than necessary, that would make k larger. For example, if instead of adding 2 to each exponent, we add 5 to the exponent of 2, making it 4 + 5 = 9, which is 3×3. But then k would have 2^5, which is 32, which is bigger than 2²=4. So, that would make k larger.

Alternatively, maybe combining exponents?

Wait, but each prime is independent. So, the exponents for different primes don't affect each other.

Therefore, the minimal exponents for each prime are 2, 2, 2.

Therefore, k is 2² × 3² × 5² = 4 × 9 × 25.

Let me compute that again.

4 × 9 is 36. 36 × 25.

Well, 36 × 25 is 36 × 100 / 4 = 900.

So, k is 900.

But wait, let me check if there's a smaller k.

Suppose I think about the cube root of 240k.

But maybe there's another way.

Alternatively, think of the exponents.

Original exponents: 2^4, 3^1, 5^1.

To make them all multiples of 3, we need to adjust each exponent.

For 2^4: 4 divided by 3 is 1 with remainder 1. So, need to add 2 to make it 6.

For 3^1: 1 divided by 3 is 0 with remainder 1. So, need to add 2 to make it 3.

Same for 5^1.

Therefore, the minimal k is 2^2 * 3^2 * 5^2.

So, 900.

But let me check if there's a smaller k.

Suppose, for example, that k has some other factors?

Wait, but if we introduce another prime factor, say 7, then we would have to have 7^3 in the product, which would make k have 7^3, which is 343, which is bigger than 900? No, 343 is smaller than 900. Wait, but if we introduce another prime, then k would have to have that prime raised to 3, but then the total k would be 2^2 * 3^2 * 5^2 * 7^3, which is way bigger than 900. So, that's worse.

Alternatively, maybe there's a way to have exponents that are multiples of 3 by combining?

Wait, no. Each prime factor is independent.

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

Original prime factors of 240: 2^4 * 3^1 * 5^1.

To make each exponent a multiple of 3, we need:

For 2: 4 + a = 3m. The smallest m such that 3m ≥ 4 is m=2, so 3*2=6. Therefore, a=2.

For 3: 1 + b = 3n. The smallest n such that 3n ≥ 1 is n=1, so 3*1=3. Therefore, b=2.

Same for 5: c=2.

Therefore, k=2^2 * 3^2 * 5^2.

So, 4*9*25=900.

But let me check with actual numbers.

Let me compute 240 * 900.

240 * 900. Well, 240 * 900 = 240 * 9 * 100 = 2160 * 100 = 216,000.

Is 216,000 a perfect cube? Let me check.

Cube root of 216,000.

Well, 60³ is 216,000. Because 6³ is 216, and 10³ is 1000, so 6*10=60, so 60³=216,000.

Yes! So, 240 * 900 = 60³.

So, that works.

But is there a smaller k? Let me think.

Suppose I try k= 2^2 * 3^2 * 5^2 / something? Wait, but k has to be an integer.

Alternatively, maybe there's a different combination?

Wait, for example, if I take exponents for 2, 3, 5 as follows:

For 2: 4 + a = 3m.

If m=1, then 3m=3, but 4 + a = 3 ⇒ a = -1, which is invalid.

So, m must be at least 2, so 6.

Similarly for 3 and 5.

So, there's no way to have lower exponents.

Alternatively, maybe if we take higher exponents for some primes and lower for others?

But since each prime is independent, you can't lower one exponent without increasing another.

Wait, for example, if I take for prime 2, instead of adding 2, maybe add 5? But that would make the exponent 4 + 5 = 9, which is 3*3. But then, for the other primes, maybe we can have lower exponents?

But no, because for 3 and 5, we still need to add 2.

So, if I take k as 2^5 * 3^2 * 5^2, that would be 32 * 9 * 25 = 32 * 225 = 7200, which is way bigger than 900.

So, that's worse.

Alternatively, if I take k as 2^2 * 3^2 * 5^something?

But for 5, we need to add 2.

Alternatively, maybe there's a different way?

Wait, maybe I made a mistake in thinking that all exponents need to be multiples of 3. Let me confirm.

Yes, because a perfect cube has exponents that are multiples of 3 in its prime factorization.

So, for example, 2^6 * 3^3 * 5^3 is (2^2 * 3 * 5)^3.

So, that's correct.

Alternatively, maybe there's a different combination where exponents are higher multiples?

But that would make k larger.

Alternatively, maybe there's a different prime factorization?

Wait, but 240 only has primes 2, 3, 5.

So, k can't have other primes unless we introduce them, which would make k larger.

Therefore, I think 900 is indeed the minimal k.

But let me check with smaller numbers.

Suppose I try k= 2^2 * 3^2 * 5^2 = 4*9*25=900.

Is there a smaller k? Let me think.

Suppose I try k= 2^2 * 3^2 * 5^1. Then, the exponents would be 2^(4+2)=6, 3^(1+2)=3, 5^(1+1)=2.

But 2 is not a multiple of 3, so that's not a cube.

Similarly, if I take k= 2^2 * 3^1 * 5^1, then exponents would be 6, 2, 2. Still not multiples of 3.

If I take k= 2^2 * 3^2 * 5^0, then exponents are 6, 3, 1. Still not.

If I take k= 2^2 * 3^0 * 5^2, same problem.

If I take k= 2^2 * 3^2 * 5^2, then exponents are 6, 3, 3. Perfect.

Alternatively, if I take k= 2^5 * 3^2 * 5^2, which is larger.

Alternatively, maybe k= 2^2 * 3^2 * 5^2 divided by something? But k has to be integer.

Alternatively, maybe there's a different combination where exponents are adjusted differently?

Wait, for example, if I take for prime 2, instead of adding 2, maybe add 5? But that would make exponent 9, which is 3*3. Then, for prime 3, maybe add 2, and for prime 5, add 2. Then, k would be 2^5 * 3^2 * 5^2. Which is 32 * 9 * 25 = 7200. Which is bigger than 900.

Alternatively, if I take for prime 2, add 2, for prime 3, add 2, and for prime 5, add 2. Which is 900.

Alternatively, if I take for prime 2, add 2, for prime 3, add 5, and for prime 5, add 5. Then, k would be 2^2 * 3^5 * 5^5. Which is way bigger.

So, no, that's worse.

Alternatively, maybe there's a way to have exponents that are multiples of 3 by combining?

Wait, for example, if I take k= 2^2 * 3^2 * 5^2, which is 900.

Alternatively, maybe there's a smaller k if we consider that some exponents can be adjusted in another way?

Wait, for example, if I take k= 2^2 * 3^2 * 5^2 / something? But k must be integer.

Alternatively, maybe there's a mistake in my initial prime factorization? Let me check again.

240 divided by 2 is 120.

120 divided by 2 is 60.

60 divided by 2 is 30.

30 divided by 2 is 15.

15 divided by 3 is 5.

5 divided by 5 is 1.

So, 2^4 * 3^1 * 5^1. Correct.

So, no mistake there.

Alternatively, maybe I can think of the problem as follows:

The product 240k is a perfect cube. So, 240k = m³ for some integer m.

Therefore, k = m³ / 240.

We need k to be an integer, so 240 must divide m³.

Therefore, m³ must be a multiple of 240.

So, m must be a multiple of the prime factors of 240.

But since 240 = 2^4 * 3^1 * 5^1, then m must be a multiple of 2^a * 3^b * 5^c, where a, b, c are such that when raised to the third power, they cover the exponents in 240.

Wait, maybe this approach is more complicated.

Alternatively, think of m as having prime factors such that when you cube it, it includes 2^4 * 3^1 * 5^1.

So, m must have exponents such that 3 times the exponent of 2 in m is at least 4, 3 times exponent of 3 in m is at least 1, and same for 5.

But since m³ must be divisible by 2^4 * 3^1 * 5^1, then:

For prime 2: 3 * exponent of 2 in m ≥ 4 ⇒ exponent of 2 in m ≥ ceil(4/3) = 2.

Similarly, for prime 3: 3 * exponent of 3 in m ≥ 1 ⇒ exponent of 3 in m ≥ 1.

For prime 5: same as 3, exponent of 5 in m ≥ 1.

Therefore, the minimal exponents for m are 2 for 2, 1 for 3, 1 for 5.

Therefore, m is 2² * 3¹ * 5¹ = 4 * 3 * 5 = 60.

Therefore, m³ = 60³ = 216000.

Then, k = m³ / 240 = 216000 / 240.

Let me compute that.

Divide numerator and denominator by 10: 21600 / 24.

Divide numerator and denominator by 12: 21600 / 12 = 1800; 24 / 12 = 2.

So, 1800 / 2 = 900.

So, k = 900.

Same result as before.

Therefore, this confirms that k is 900.

But let me check if there is a smaller m.

Suppose m is smaller. For example, if m is 2² * 3¹ * 5¹ = 60.

If I try m = 2² * 3¹ * 5¹ * something?

But if m is smaller, then m³ would be smaller, but then k would be smaller? Wait, no. Because if m is smaller, then m³ is smaller, but k = m³ / 240.

But if m is smaller, then m³ might not be divisible by 240.

Wait, for example, if m is 2² * 3¹ * 5⁰ = 4 * 3 = 12. Then m³ = 12³ = 1728.

Then, k = 1728 / 240 = 7.2. Not integer.

So, invalid.

If m is 2² * 3¹ * 5¹ = 60, then k is 900.

If m is 2³ * 3¹ * 5¹ = 8 * 3 * 5 = 120. Then m³ = 120³.

k = 120³ / 240 = (120 / 240) * 120² = (1/2) * 14400 = 7200. Which is bigger than 900.

So, worse.

Alternatively, m = 2² * 3² * 5¹. Then m³ = (2² * 3² * 5¹)^3 = 2^6 * 3^6 * 5^3.

Then, k = m³ / 240 = (2^6 * 3^6 * 5^3) / (2^4 * 3^1 * 5^1) ) = 2^(6-4) * 3^(6-1) * 5^(3-1) = 2² * 3^5 * 5². Which is 4 * 243 * 25 = 4 * 6075 = 24300. Which is way bigger.

So, no.

Alternatively, m = 2² * 3¹ * 5². Then m³ = 2^6 * 3^3 * 5^6. Then k = m³ / 240 = 2^(6-4) * 3^(3-1) * 5^(6-1) = 2² * 3² * 5^5. Which is 4 * 9 * 3125 = 112500. Also bigger.

So, no.

Therefore, the minimal m is 60, leading to k=900.

Therefore, I think 900 is indeed the minimal k.

But let me check if there's another m with different exponents.

For example, suppose m has exponents for 2 as 2, for 3 as 1, for 5 as 1. Which is 60.

Alternatively, if m has exponents for 2 as 2, for 3 as 1, for 5 as 1, and maybe another prime?

But then m³ would have that prime cubed, so k would have that prime cubed divided by 240. But since 240 doesn't have that prime, k would have that prime cubed. Which would make k larger.

So, no.

Alternatively, if m has exponents for 2 as 2, for 3 as 1, for 5 as 1, and maybe some other exponents?

But again, same problem.

Therefore, I think 900 is the minimal k.

But just to be thorough, let me check if there's a k smaller than 900.

Suppose k is 180. Let me check if 240 * 180 is a perfect cube.

240 * 180 = 43200.

What's the cube root of 43200?

Well, 35³ is 42875, 36³ is 46656. So, between 35 and 36. Not an integer.

Not a perfect cube.

How about k= 240? 240*240=57600. Cube root? 38³=54872, 39³=59319. Still not.

k= 360. 240*360=86400. Cube root? 44³=85184, 45³=91125. Not a cube.

k= 450. 240*450=108000. Cube root of 108000. 47³=103823, 48³=110592. Not a cube.

k= 600. 240*600=144000. Cube root of 144000. 52³=140608, 53³=148877. Not a cube.

k= 720. 240*720=172800. Cube root? 55³=166375, 56³=175616. Not a cube.

k= 900. 240*900=216000. Which is 60³.

So, yes, 60³.

So, seems like 900 is the first one that works.

Alternatively, maybe k= 2^2 * 3^2 * 5^2 / something? But k must be integer.

Alternatively, maybe there's a k with exponents less than 2 for some primes?

But we saw that for each prime, the exponents need to be at least 2.

For example, if I try k= 2^2 * 3^2 * 5^1. Then, 240 * k = 2^6 * 3^3 * 5^2. The exponent of 5 is 2, which is not a multiple of 3. So, not a cube.

Similarly, if I take k= 2^2 * 3^1 * 5^2, same problem with 3.

If I take k= 2^1 * 3^2 * 5^2, then exponent of 2 is 4 + 1 = 5, not multiple of 3.

So, no.

Therefore, all exponents need to be increased by 2.

Therefore, k= 2² * 3² * 5²= 900.

Therefore, I think that is the minimal k.

So, the answer is 900.

**Final Answer**
The smallest possible positive value of $ k $ is \boxed{900}.

To solve the problem, we are given that the product of the integers 240 and $ k $ is a **perfect cube**, and we are to find the **smallest possible positive value** of $ k $.

---

### Step 1: Prime Factorization of 240

We begin by factoring 240 into its prime components:

$$
240 = 2^4 \times 3^1 \times 5^1
$$

---

### Step 2: Understanding the Condition for a Perfect Cube

A number is a **perfect cube** if all the exponents in its prime factorization are **multiples of 3**. So, for the product $ 240 \times k $ to be a perfect cube, the combined exponents of each prime factor in the product must be divisible by 3.

Let’s denote the prime factorization of $ k $ as:

$$
k = 2^a \times 3^b \times 5^c
$$

Then the product becomes:

$$
240 \times k = 2^{4+a} \times 3^{1+b} \times 5^{1+c}
$$

We now require:

- $ 4 + a \equiv 0 \pmod{3} $
- $ 1 + b \equiv 0 \pmod{3} $
- $ 1 + c \equiv 0 \pmod{3} $

---

### Step 3: Solve for the Smallest Exponents

We solve each congruence for the smallest non-negative integer values of $ a, b, c $:

- $ 4 + a \equiv 0 \pmod{3} \Rightarrow a = 2 $
- $ 1 + b \equiv 0 \pmod{3} \Rightarrow b = 2 $
- $ 1 + c \equiv 0 \pmod{3} \Rightarrow c = 2 $

So, the smallest such $ k $ is:

$$
k = 2^2 \times 3^2 \times 5^2 = 4 \times 9 \times 25 = 900
$$

---

### Step 4: Verification

Let’s verify that $ 240 \times 900 $ is indeed a perfect cube:

$$
240 \times 900 = 216000
$$

Now, compute the cube root:

$$
\sqrt[3]{216000} = 60
$$

So, $ 216000 = 60^3 $, which is a perfect cube.

---

### Step 5: Confirm Minimality

We can confirm that no smaller $ k $ will satisfy the condition. Any smaller value would result in at least one exponent in the product not being divisible by 3, which would prevent the product from being a perfect cube.

---

### Final Answer

$$
\boxed{900}
$$

[Assistant]
Explanation:
Okay, so I need to find the smallest positive integer k such that when I multiply 240 by k, the result is a perfect cube. Hmm, let me think about how to approach this.

First, I remember that a perfect cube is a number that can be expressed as some integer raised to the power of three. For example, 8 is a perfect cube because it's 2³, and 27 is 3³. So, if 240 times k is a perfect cube, then all the prime factors in its prime factorization must have exponents that are multiples of three.

Alright, so maybe I should start by factoring 240 into its prime factors. Let me do that.

Breaking down 240:

240 divided by 2 is 120.
120 divided by 2 is 60.
60 divided by 2 is 30.
30 divided by 2 is 15.
15 divided by 3 is 5.
5 divided by 5 is 1.

So, the prime factors of 240 are 2⁴ × 3¹ × 5¹. Let me check that again.

Yes, 2⁴ is 16, 3¹ is 3, 5¹ is 5. Multiplying them together: 16 × 3 = 48, 48 × 5 = 240. Perfect.

So, 240 = 2⁴ × 3¹ × 5¹.

Now, if I multiply this by k, the product should be a perfect cube. Let me denote the prime factorization of k as something. Let me think.

Suppose k has prime factors 2^a × 3^b × 5^c × ... (other primes if necessary). Then, multiplying by 240 would give:

(2⁴ × 3¹ × 5¹) × (2^a × 3^b × 5^c × ...) = 2^(4+a) × 3^(1+b) × 5^(1+c) × ...

For this product to be a perfect cube, each exponent in the prime factorization must be a multiple of 3.

So, for each prime, the exponent after multiplying by k must be divisible by 3.

So, let's consider each prime factor:

For prime 2: exponent is 4 + a. This needs to be a multiple of 3.

Similarly, for prime 3: exponent is 1 + b. Needs to be multiple of 3.

For prime 5: exponent is 1 + c. Needs to be multiple of 3.

And for any other primes in k, their exponents must also be multiples of 3.

But since we want the smallest possible k, we should avoid introducing any new primes. Because adding new primes would require their exponents to be multiples of 3, which would make k larger. So, probably k only needs to have primes 2, 3, and 5.

So, let me focus on those.

Starting with prime 2: 4 + a ≡ 0 mod 3.

So, 4 + a must be divisible by 3. Let me solve for a.

What's the smallest non-negative integer a such that 4 + a is divisible by 3?

Well, 4 divided by 3 leaves a remainder of 1. So, to make it divisible by 3, we need to add 2, because 1 + 2 = 3. So, a = 2.

Wait, let me check: 4 + 2 = 6, which is 3×2. Yes, that works.

Alternatively, if a = 2, then exponent of 2 becomes 6, which is 3×2.

But is there a smaller a? Well, a has to be non-negative. If a = -1, that would be 3, but negative exponents aren't allowed because k is a positive integer. So, the smallest a is 2.

Okay, moving on to prime 3: exponent is 1 + b.

Similarly, 1 + b must be divisible by 3.

So, 1 + b ≡ 0 mod 3 ⇒ b ≡ -1 mod 3 ⇒ b ≡ 2 mod 3.

So, the smallest non-negative integer b is 2.

Same with prime 5: exponent is 1 + c. So, same as prime 3. So, c = 2.

Therefore, k would be 2² × 3² × 5².

Wait, let me check that.

So, if k is 2² × 3² × 5², then multiplying by 240:

Original exponents: 2⁴ × 3¹ × 5¹.

After multiplying by k: 2^(4+2) × 3^(1+2) × 5^(1+2) = 2^6 × 3^3 × 5^3.

Which is (2² × 3 × 5)^3.

Yes, because 2^6 is (2²)^3, 3^3 is (3)^3, 5^3 is (5)^3. So, altogether, it's (2² × 3 × 5)^3.

So, that's a perfect cube.

But wait, is this the smallest possible k? Let me check if there's a smaller k.

Wait, maybe I can have exponents that are higher multiples? For example, for prime 2, instead of adding 2 to get 6, could I add 5 to get 9? But that would make k larger. Similarly for others. So, the minimal exponents are 2 for each.

But let me think again.

Alternatively, maybe there's a way to have exponents that are multiples of 3 by subtracting? But since exponents can't be negative, we can't subtract.

So, for each prime, we need to find the minimal exponent to add to make the total exponent a multiple of 3.

So, for prime 2: original exponent is 4. The next multiple of 3 after 4 is 6. So, need to add 2.

For prime 3: original exponent is 1. Next multiple of 3 is 3. So, need to add 2.

Same for prime 5: original exponent is 1. Next multiple of 3 is 3. So, need to add 2.

Therefore, k is 2² × 3² × 5².

Calculating that: 2² is 4, 3² is 9, 5² is 25.

So, 4 × 9 = 36, 36 × 25 = 900.

So, k is 900?

Wait, but let me check if there's a smaller k.

Is there a possibility that some exponents can be adjusted in a way that k is smaller?

For example, maybe if we have some exponents that are higher multiples?

Wait, but if we take exponents higher than necessary, that would make k larger. For example, if instead of adding 2 to each exponent, we add 5 to the exponent of 2, making it 4 + 5 = 9, which is 3×3. But then k would have 2^5, which is 32, which is bigger than 2²=4. So, that would make k larger.

Alternatively, maybe combining exponents?

Wait, but each prime is independent. So, the exponents for different primes don't affect each other.

Therefore, the minimal exponents for each prime are 2, 2, 2.

Therefore, k is 2² × 3² × 5² = 4 × 9 × 25.

Let me compute that again.

4 × 9 is 36. 36 × 25.

Well, 36 × 25 is 36 × 100 / 4 = 900.

So, k is 900.

But wait, let me check if there's a smaller k.

Suppose I think about the cube root of 240k.

But maybe there's another way.

Alternatively, think of the exponents.

Original exponents: 2^4, 3^1, 5^1.

To make them all multiples of 3, we need to adjust each exponent.

For 2^4: 4 divided by 3 is 1 with remainder 1. So, need to add 2 to make it 6.

For 3^1: 1 divided by 3 is 0 with remainder 1. So, need to add 2 to make it 3.

Same for 5^1.

Therefore, the minimal k is 2^2 * 3^2 * 5^2.

So, 900.

But let me check if there's a smaller k.

Suppose, for example, that k has some other factors?

Wait, but if we introduce another prime factor, say 7, then we would have to have 7^3 in the product, which would make k have 7^3, which is 343, which is bigger than 900? No, 343 is smaller than 900. Wait, but if we introduce another prime, then k would have to have that prime raised to 3, but then the total k would be 2^2 * 3^2 * 5^2 * 7^3, which is way bigger than 900. So, that's worse.

Alternatively, maybe there's a way to have exponents that are multiples of 3 by combining?

Wait, no. Each prime factor is independent.

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

Original prime factors of 240: 2^4 * 3^1 * 5^1.

To make each exponent a multiple of 3, we need:

For 2: 4 + a = 3m. The smallest m such that 3m ≥ 4 is m=2, so 3*2=6. Therefore, a=2.

For 3: 1 + b = 3n. The smallest n such that 3n ≥ 1 is n=1, so 3*1=3. Therefore, b=2.

Same for 5: c=2.

Therefore, k=2^2 * 3^2 * 5^2.

So, 4*9*25=900.

But let me check with actual numbers.

Let me compute 240 * 900.

240 * 900. Well, 240 * 900 = 240 * 9 * 100 = 2160 * 100 = 216,000.

Is 216,000 a perfect cube? Let me check.

Cube root of 216,000.

Well, 60³ is 216,000. Because 6³ is 216, and 10³ is 1000, so 6*10=60, so 60³=216,000.

Yes! So, 240 * 900 = 60³.

So, that works.

But is there a smaller k? Let me think.

Suppose I try k= 2^2 * 3^2 * 5^2 / something? Wait, but k has to be an integer.

Alternatively, maybe there's a different combination?

Wait, for example, if I take exponents for 2, 3, 5 as follows:

For 2: 4 + a = 3m.

If m=1, then 3m=3, but 4 + a = 3 ⇒ a = -1, which is invalid.

So, m must be at least 2, so 6.

Similarly for 3 and 5.

So, there's no way to have lower exponents.

Alternatively, maybe if we take higher exponents for some primes and lower for others?

But since each prime is independent, you can't lower one exponent without increasing another.

Wait, for example, if I take for prime 2, instead of adding 2, maybe add 5? But that would make the exponent 4 + 5 = 9, which is 3*3. But then, for the other primes, maybe we can have lower exponents?

But no, because for 3 and 5, we still need to add 2.

So, if I take k as 2^5 * 3^2 * 5^2, that would be 32 * 9 * 25 = 32 * 225 = 7200, which is way bigger than 900.

So, that's worse.

Alternatively, if I take k as 2^2 * 3^2 * 5^something?

But for 5, we need to add 2.

Alternatively, maybe there's a different way?

Wait, maybe I made a mistake in thinking that all exponents need to be multiples of 3. Let me confirm.

Yes, because a perfect cube has exponents that are multiples of 3 in its prime factorization.

So, for example, 2^6 * 3^3 * 5^3 is (2^2 * 3 * 5)^3.

So, that's correct.

Alternatively, maybe there's a different combination where exponents are higher multiples?

But that would make k larger.

Alternatively, maybe there's a different prime factorization?

Wait, but 240 only has primes 2, 3, 5.

So, k can't have other primes unless we introduce them, which would make k larger.

Therefore, I think 900 is indeed the minimal k.

But let me check with smaller numbers.

Suppose I try k= 2^2 * 3^2 * 5^2 = 4*9*25=900.

Is there a smaller k? Let me think.

Suppose I try k= 2^2 * 3^2 * 5^1. Then, the exponents would be 2^(4+2)=6, 3^(1+2)=3, 5^(1+1)=2.

But 2 is not a multiple of 3, so that's not a cube.

Similarly, if I take k= 2^2 * 3^1 * 5^1, then exponents would be 6, 2, 2. Still not multiples of 3.

If I take k= 2^2 * 3^2 * 5^0, then exponents are 6, 3, 1. Still not.

If I take k= 2^2 * 3^0 * 5^2, same problem.

If I take k= 2^2 * 3^2 * 5^2, then exponents are 6, 3, 3. Perfect.

Alternatively, if I take k= 2^5 * 3^2 * 5^2, which is larger.

Alternatively, maybe k= 2^2 * 3^2 * 5^2 divided by something? But k has to be integer.

Alternatively, maybe there's a different combination where exponents are adjusted differently?

Wait, for example, if I take for prime 2, instead of adding 2, maybe add 5? But that would make exponent 9, which is 3*3. Then, for prime 3, maybe add 2, and for prime 5, add 2. Then, k would be 2^5 * 3^2 * 5^2. Which is 32 * 9 * 25 = 7200. Which is bigger than 900.

Alternatively, if I take for prime 2, add 2, for prime 3, add 2, and for prime 5, add 2. Which is 900.

Alternatively, if I take for prime 2, add 2, for prime 3, add 5, and for prime 5, add 5. Then, k would be 2^2 * 3^5 * 5^5. Which is way bigger.

So, no, that's worse.

Alternatively, maybe there's a way to have exponents that are multiples of 3 by combining?

Wait, for example, if I take k= 2^2 * 3^2 * 5^2, which is 900.

Alternatively, maybe there's a smaller k if we consider that some exponents can be adjusted in another way?

Wait, for example, if I take k= 2^2 * 3^2 * 5^2 / something? But k must be integer.

Alternatively, maybe there's a mistake in my initial prime factorization? Let me check again.

240 divided by 2 is 120.

120 divided by 2 is 60.

60 divided by 2 is 30.

30 divided by 2 is 15.

15 divided by 3 is 5.

5 divided by 5 is 1.

So, 2^4 * 3^1 * 5^1. Correct.

So, no mistake there.

Alternatively, maybe I can think of the problem as follows:

The product 240k is a perfect cube. So, 240k = m³ for some integer m.

Therefore, k = m³ / 240.

We need k to be an integer, so 240 must divide m³.

Therefore, m³ must be a multiple of 240.

So, m must be a multiple of the prime factors of 240.

But since 240 = 2^4 * 3^1 * 5^1, then m must be a multiple of 2^a * 3^b * 5^c, where a, b, c are such that when raised to the third power, they cover the exponents in 240.

Wait, maybe this approach is more complicated.

Alternatively, think of m as having prime factors such that when you cube it, it includes 2^4 * 3^1 * 5^1.

So, m must have exponents such that 3 times the exponent of 2 in m is at least 4, 3 times exponent of 3 in m is at least 1, and same for 5.

But since m³ must be divisible by 2^4 * 3^1 * 5^1, then:

For prime 2: 3 * exponent of 2 in m ≥ 4 ⇒ exponent of 2 in m ≥ ceil(4/3) = 2.

Similarly, for prime 3: 3 * exponent of 3 in m ≥ 1 ⇒ exponent of 3 in m ≥ 1.

For prime 5: same as 3, exponent of 5 in m ≥ 1.

Therefore, the minimal exponents for m are 2 for 2, 1 for 3, 1 for 5.

Therefore, m is 2² * 3¹ * 5¹ = 4 * 3 * 5 = 60.

Therefore, m³ = 60³ = 216000.

Then, k = m³ / 240 = 216000 / 240.

Let me compute that.

Divide numerator and denominator by 10: 21600 / 24.

Divide numerator and denominator by 12: 21600 / 12 = 1800; 24 / 12 = 2.

So, 1800 / 2 = 900.

So, k = 900.

Same result as before.

Therefore, this confirms that k is 900.

But let me check if there is a smaller m.

Suppose m is smaller. For example, if m is 2² * 3¹ * 5¹ = 60.

If I try m = 2² * 3¹ * 5¹ * something?

But if m is smaller, then m³ would be smaller, but then k would be smaller? Wait, no. Because if m is smaller, then m³ is smaller, but k = m³ / 240.

But if m is smaller, then m³ might not be divisible by 240.

Wait, for example, if m is 2² * 3¹ * 5⁰ = 4 * 3 = 12. Then m³ = 12³ = 1728.

Then, k = 1728 / 240 = 7.2. Not integer.

So, invalid.

If m is 2² * 3¹ * 5¹ = 60, then k is 900.

If m is 2³ * 3¹ * 5¹ = 8 * 3 * 5 = 120. Then m³ = 120³.

k = 120³ / 240 = (120 / 240) * 120² = (1/2) * 14400 = 7200. Which is bigger than 900.

So, worse.

Alternatively, m = 2² * 3² * 5¹. Then m³ = (2² * 3² * 5¹)^3 = 2^6 * 3^6 * 5^3.

Then, k = m³ / 240 = (2^6 * 3^6 * 5^3) / (2^4 * 3^1 * 5^1) ) = 2^(6-4) * 3^(6-1) * 5^(3-1) = 2² * 3^5 * 5². Which is 4 * 243 * 25 = 4 * 6075 = 24300. Which is way bigger.

So, no.

Alternatively, m = 2² * 3¹ * 5². Then m³ = 2^6 * 3^3 * 5^6. Then k = m³ / 240 = 2^(6-4) * 3^(3-1) * 5^(6-1) = 2² * 3² * 5^5. Which is 4 * 9 * 3125 = 112500. Also bigger.

So, no.

Therefore, the minimal m is 60, leading to k=900.

Therefore, I think 900 is indeed the minimal k.

But let me check if there's another m with different exponents.

For example, suppose m has exponents for 2 as 2, for 3 as 1, for 5 as 1. Which is 60.

Alternatively, if m has exponents for 2 as 2, for 3 as 1, for 5 as 1, and maybe another prime?

But then m³ would have that prime cubed, so k would have that prime cubed divided by 240. But since 240 doesn't have that prime, k would have that prime cubed. Which would make k larger.

So, no.

Alternatively, if m has exponents for 2 as 2, for 3 as 1, for 5 as 1, and maybe some other exponents?

But again, same problem.

Therefore, I think 900 is the minimal k.

But just to be thorough, let me check if there's a k smaller than 900.

Suppose k is 180. Let me check if 240 * 180 is a perfect cube.

240 * 180 = 43200.

What's the cube root of 43200?

Well, 35³ is 42875, 36³ is 46656. So, between 35 and 36. Not an integer.

Not a perfect cube.

How about k= 240? 240*240=57600. Cube root? 38³=54872, 39³=59319. Still not.

k= 360. 240*360=86400. Cube root? 44³=85184, 45³=91125. Not a cube.

k= 450. 240*450=108000. Cube root of 108000. 47³=103823, 48³=110592. Not a cube.

k= 600. 240*600=144000. Cube root of 144000. 52³=140608, 53³=148877. Not a cube.

k= 720. 240*720=172800. Cube root? 55³=166375, 56³=175616. Not a cube.

k= 900. 240*900=216000. Which is 60³.

So, yes, 60³.

So, seems like 900 is the first one that works.

Alternatively, maybe k= 2^2 * 3^2 * 5^2 / something? But k must be integer.

Alternatively, maybe there's a k with exponents less than 2 for some primes?

But we saw that for each prime, the exponents need to be at least 2.

For example, if I try k= 2^2 * 3^2 * 5^1. Then, 240 * k = 2^6 * 3^3 * 5^2. The exponent of 5 is 2, which is not a multiple of 3. So, not a cube.

Similarly, if I take k= 2^2 * 3^1 * 5^2, same problem with 3.

If I take k= 2^1 * 3^2 * 5^2, then exponent of 2 is 4 + 1 = 5, not multiple of 3.

So, no.

Therefore, all exponents need to be increased by 2.

Therefore, k= 2² * 3² * 5²= 900.

Therefore, I think that is the minimal k.

So, the answer is 900.

**Final Answer**
The smallest possible positive value of $ k $ is \boxed{900}.

To solve the problem, we are given that the product of the integers 240 and $ k $ is a **perfect cube**, and we are to find the **smallest possible positive value** of $ k $.

---

### Step 1: Prime Factorization of 240

We begin by factoring 240 into its prime components:

$$
240 = 2^4 \times 3^1 \times 5^1
$$

---

### Step 2: Understanding the Condition for a Perfect Cube

A number is a **perfect cube** if all the exponents in its prime factorization are **multiples of 3**. So, for the product $ 240 \times k $ to be a perfect cube, the combined exponents of each prime factor in the product must be divisible by 3.

Let’s denote the prime factorization of $ k $ as:

$$
k = 2^a \times 3^b \times 5^c
$$

Then the product becomes:

$$
240 \times k = 2^{4+a} \times 3^{1+b} \times 5^{1+c}
$$

We now require:

- $ 4 + a \equiv 0 \pmod{3} $
- $ 1 + b \equiv 0 \pmod{3} $
- $ 1 + c \equiv 0 \pmod{3} $

---

### Step 3: Solve for the Smallest Exponents

We solve each congruence for the smallest non-negative integer values of $ a, b, c $:

- $ 4 + a \equiv 0 \pmod{3} \Rightarrow a = 2 $
- $ 1 + b \equiv 0 \pmod{3} \Rightarrow b = 2 $
- $ 1 + c \equiv 0 \pmod{3} \Rightarrow c = 2 $

So, the smallest such $ k $ is:

$$
k = 2^2 \times 3^2 \times 5^2 = 4 \times 9 \times 25 = 900
$$

---

### Step 4: Verification

Let’s verify that $ 240 \times 900 $ is indeed a perfect cube:

$$
240 \times 900 = 216000
$$

Now, compute the cube root:

$$
\sqrt[3]{216000} = 60
$$

So, $ 216000 = 60^3 $, which is a perfect cube.

---

### Step 5: Confirm Minimality

We can confirm that no smaller $ k $ will satisfy the condition. Any smaller value would result in at least one exponent in the product not being divisible by 3, which would prevent the product from being a perfect cube.

---

### Final Answer

$$
\boxed{900}
$$

Solution: $240=2^4\cdot3\cdot5=2^3(2\cdot3\cdot5)$. For $240k$ to be a perfect cube (and not a perfect square), $k$ must be at least $2^2\cdot3^2\cdot5^2=\boxed{900}$.


Explanation:
Okay, so I need to find the sum of the digits in the terminating decimal representation of the fraction 4321 divided by (5^7 times 2^8). Hmm, let me think about how to approach this.

First, I remember that when you have a fraction with denominators that are powers of 2 and/or 5, the decimal will terminate. That's because 2 and 5 are the prime factors of 10, the base of our number system. So, the decimal will terminate, and the number of decimal places will be determined by the maximum exponent between the powers of 2 and 5 in the denominator.

Wait, let me recall the exact rule. If you have a fraction a/b where b is of the form 2^n * 5^m, then the decimal terminates after max(n, m) digits. So, in this case, the denominator is 5^7 * 2^8. Therefore, the maximum of 7 and 8 is 8. So, the decimal will terminate after 8 digits. That means if I convert this fraction to a decimal, it will have 8 digits after the decimal point, and then it will end.

But how do I actually compute the decimal? Well, one way is to multiply numerator and denominator by something to make the denominator a power of 10. Since the denominator is 5^7 * 2^8, if I multiply numerator and denominator by 5^(something) to make the denominator 10^8? Wait, let me check.

The denominator is 5^7 * 2^8. If I want to make it 10^k, then I need to have both 2^k and 5^k. Since the denominator already has 2^8 and 5^7, to make it 10^8, I need to multiply by 5^(8 - 7) = 5^1. Because 5^7 * 5^1 = 5^8, and 2^8 * 5^1 = 2^8 * 5^1. Wait, no, that would make the denominator 2^8 * 5^8, which is 10^8. So, if I multiply numerator and denominator by 5^1, then the denominator becomes 10^8, and the numerator becomes 4321 * 5.

Let me write that down:

Original fraction: 4321 / (5^7 * 2^8)

Multiply numerator and denominator by 5^1:

Numerator: 4321 * 5

Denominator: 5^7 * 2^8 * 5^1 = 5^(7+1) * 2^8 = 5^8 * 2^8 = (5*2)^8 = 10^8

So, the fraction becomes (4321 * 5) / 10^8. Therefore, the decimal representation is (4321 * 5) divided by 10^8, which is the same as moving the decimal point 8 places to the left in the numerator.

So, first, let me compute 4321 * 5. Let me do that multiplication:

4321 * 5:

Well, 4000 * 5 = 20,000

300 * 5 = 1,500

20 * 5 = 100

1 * 5 = 5

Adding them up: 20,000 + 1,500 = 21,500; 21,500 + 100 = 21,600; 21,600 + 5 = 21,605.

So, 4321 * 5 = 21,605.

Therefore, the fraction is 21,605 / 10^8.

So, 21,605 divided by 10^8 is 0.000021605. Wait, let me check that.

Dividing by 10^8 moves the decimal point 8 places to the left. Let me write 21,605 as 21605.0.

Starting from the right, moving decimal 8 places left:

Original number: 21605.0

Move decimal 1 place left: 2160.50

2 places: 216.050

3 places: 21.6050

4 places: 2.16050

5 places: 0.216050

6 places: 0.0216050

7 places: 0.00216050

8 places: 0.000216050

Wait, so that would be 0.00021605? But wait, the numerator is 21,605. So, 21,605 divided by 10^8 is 0.00021605? Let me check with another approach.

Alternatively, 10^8 is 100,000,000. So, 21,605 divided by 100,000,000.

So, 21,605 / 100,000,000 = 0.00021605.

Yes, because 21,605 is 2.1605 x 10^4, so divided by 10^8 is 2.1605 x 10^-4, which is 0.00021605.

But wait, let me count the decimal places. Let me write 21,605 as 21605. So, to write it as a decimal with 8 decimal places, we need to add leading zeros.

So, 21605 is the same as 000021605. If we have 8 decimal places, then it's 0.000021605? Wait, hold on. Wait, confusion here.

Wait, if I have 21,605 divided by 10^8, then it's equal to 21,605 divided by 100,000,000. Let me write that as a decimal.

Let me think of 21,605 divided by 100,000,000.

Dividing by 10^8 is moving the decimal point 8 places to the left.

Starting with 21,605.0

First decimal place: 2160.50

Second: 216.050

Third: 21.6050

Fourth: 2.16050

Fifth: 0.216050

Sixth: 0.0216050

Seventh: 0.00216050

Eighth: 0.000216050

So, after moving the decimal 8 places, it's 0.00021605. But wait, the original number is 21,605. So, how many digits do we have? Let me think.

The number 21,605 has 5 digits. When we divide by 10^8, we need to add 8 - 5 = 3 leading zeros? Wait, no. Wait, if you have a number with n digits, dividing by 10^k will result in a decimal with k digits after the decimal point, but depending on the number.

Alternatively, think of 21,605 as 21605. So, to write this as a decimal with 8 decimal places, we need to add 3 zeros before the decimal point? Wait, no. Let me think of it as:

The decimal representation is 0. followed by (number of zeros) then the digits of the numerator.

Since 21,605 is 21605, which is 5 digits. To have 8 decimal places, we need to have 8 digits after the decimal. So, starting from the right, we need to add 3 zeros before the digits.

Wait, for example, if I have 123 divided by 10^5, that's 0.00123. So, 123 is 3 digits, divided by 10^5, so 5 decimal places: 0.00123. So, adding two zeros before the 123.

Similarly, 21,605 divided by 10^8 is 0.00021605. Let me count the decimal places:

0.00021605 has 8 decimal places. Let me count:

First decimal place: 0

Second: 0

Third: 0

Fourth: 2

Fifth: 1

Sixth: 6

Seventh: 0

Eighth: 5

Wait, but that's only 8 digits. Wait, but 21,605 is 5 digits. So, how does that work?

Wait, 21,605 divided by 10^8 is equal to 0.00021605. Let me check with actual division.

Let me write 21,605 divided by 100,000,000.

So, 21,605 ÷ 100,000,000.

Well, 100,000,000 goes into 21,605 zero times. So, 0. Then, we add decimal places.

How many decimal places? Let me think.

Alternatively, think of 21,605 as 2.1605 x 10^4. Then, dividing by 10^8 is 2.1605 x 10^(4 - 8) = 2.1605 x 10^-4 = 0.00021605.

Yes, that's correct. So, 0.00021605.

So, the decimal is 0.00021605.

But wait, let me check if there are more digits. Because 21,605 is 5 digits, and we have 8 decimal places. So, the decimal would be 0.00021605. Let me count the digits after the decimal: 0, 0, 0, 2, 1, 6, 0, 5. That's 8 digits.

So, the decimal is 0.00021605.

But wait, is that correct? Let me check with another method.

Alternatively, since the denominator is 5^7 * 2^8, which is 5^7 * 2^7 * 2^1 = (5*2)^7 * 2 = 10^7 * 2. Therefore, the denominator is 10^7 * 2.

So, the original fraction is 4321 / (10^7 * 2) = (4321 / 2) / 10^7.

So, 4321 divided by 2 is 2160.5. Then, divided by 10^7 is 0.00021605.

Yes! That's the same result. So, 0.00021605.

So, the decimal is 0.00021605.

Now, the question is asking for the sum of the digits in the terminating decimal representation.

So, first, let me write out the decimal: 0.00021605.

But wait, is that all? Let me check if there are more digits. Since the decimal terminates after 8 digits, as we determined earlier.

But let me confirm. The decimal is 0.00021605. Let me count the digits after the decimal:

Position 1: 0

Position 2: 0

Position 3: 0

Position 4: 2

Position 5: 1

Position 6: 6

Position 7: 0

Position 8: 5

So, that's 8 digits. So, the decimal is 0.00021605.

But wait, when I did the calculation earlier, I thought of it as 0.00021605. However, when I first thought of moving the decimal, I thought of 0.00021605. But let me check if there's a trailing zero or something.

Wait, 21,605 divided by 10^8 is 0.00021605. So, that's exactly 8 decimal digits.

But let me check with another approach. Let me compute 4321 divided by (5^7 * 2^8).

Alternatively, compute denominator first: 5^7 is 78125, and 2^8 is 256. So, 78125 * 256. Let me compute that.

But that might be tedious, but maybe I can do it step by step.

First, 78125 * 256. Let me note that 256 is 2^8, which is 256.

Alternatively, 78125 * 256 = 78125 * (200 + 50 + 6) = 78125*200 + 78125*50 + 78125*6.

But maybe there's a better way.

Alternatively, note that 78125 * 256 = 78125 * 2^8. Since 78125 is 5^7, so 5^7 * 2^8 = 10^7 * 2.

Wait, that's the same as before. So, denominator is 10^7 * 2.

So, 4321 divided by (10^7 * 2) is equal to (4321 / 2) / 10^7 = 2160.5 / 10^7 = 0.00021605.

Yes, same result.

So, decimal is 0.00021605.

Therefore, the digits after the decimal are 0, 0, 0, 2, 1, 6, 0, 5.

So, the digits are: 0, 0, 0, 2, 1, 6, 0, 5.

Now, the question says "the sum of the digits in the terminating decimal representation".

But wait, does that include the digits before the decimal? Well, the decimal representation is 0.00021605. The digits before the decimal are just 0, which is a single digit. But usually, when they say "sum of the digits in the decimal representation", they might be referring to the digits after the decimal point. However, sometimes they might include the integer part.

But let me check the problem statement again: "the sum of the digits in the terminating decimal representation of the fraction".

The decimal representation is 0.00021605. So, the digits are 0 (before decimal), then 0, 0, 0, 2, 1, 6, 0, 5.

But if we consider the entire decimal representation, including the integer part, then the digits are 0, 0, 0, 2, 1, 6, 0, 5. However, the integer part is 0, so adding that would be 0 + 0 + 0 + 2 + 1 + 6 + 0 + 5.

But maybe the problem is referring only to the digits after the decimal? Let me think.

In some contexts, when they say "decimal representation", they might mean the fractional part. But to be safe, let me check both possibilities.

But let me think again. The fraction is 4321/(5^7*2^8). When converted to decimal, it's 0.00021605. So, the entire decimal representation is 0.00021605. The digits are 0, 0, 0, 2, 1, 6, 0, 5.

But if we consider the entire number, including the leading zero before the decimal, then the digits are 0 (before decimal) and then the decimal digits. However, in decimal representations, leading zeros before the decimal are not considered as part of the digits in the number. For example, the number 0.00021605 is just 0.00021605, and the digits are 0, 0, 0, 2, 1, 6, 0, 5.

But if you consider the entire decimal expansion, including the integer part, then the digits are 0 (from the integer part) and then the decimal digits. However, the integer part is 0, so adding that would be 0 + 0 + 0 + 2 + 1 + 6 + 0 + 5.

But maybe the problem is referring only to the digits after the decimal point. Let me check the problem statement again: "the sum of the digits in the terminating decimal representation".

The term "decimal representation" usually includes all digits, both before and after the decimal. However, in some contexts, especially when talking about decimal expansions, people might refer to the fractional part.

But to be precise, let me check with an example. For instance, if the decimal was 0.123, the digits are 0, 1, 2, 3. But if you consider the entire number, it's 0.123, so digits are 0, 1, 2, 3. However, if you have a number like 123.456, the digits are 1, 2, 3, 4, 5, 6.

But in our case, the number is 0.00021605. So, the digits are 0, 0, 0, 2, 1, 6, 0, 5. However, the leading zero before the decimal is part of the decimal representation. But in reality, the number is 0.00021605, so the digits are 0 (before decimal), then 0, 0, 0, 2, 1, 6, 0, 5.

But if you consider the decimal representation as a string, it's "0.00021605", so the digits are 0, 0, 0, 2, 1, 6, 0, 5. However, the leading zero before the decimal is not considered as part of the decimal digits. Wait, actually, in the decimal representation, the digits are both before and after the decimal.

But in the case of 0.00021605, the digits are 0 (units place), 0 (tenths), 0 (hundredths), 0 (thousandths), 2 (ten-thousandths), 1 (hundred-thousandths), 6 (millionths), 0 (ten-millionths), 5 (hundred-millionths).

But if you consider the entire decimal representation, the digits are 0, 0, 0, 0, 2, 1, 6, 0, 5. Wait, but that would be 9 digits. Wait, confusion arises here.

Wait, the number is 0.00021605. Let me write it out with all decimal places:

Position: 1: 0 (tenths)

Position 2: 0 (hundredths)

Position 3: 0 (thousandths)

Position 4: 0 (ten-thousandths)

Position 5: 2 (hundred-thousandths)

Position 6: 1 (millionths)

Position 7: 6 (ten-millionths)

Position 8: 0 (hundred-millionths)

Position 9: 5 (billionths)

Wait, but the decimal terminates after 8 digits. Wait, earlier we said that the decimal terminates after 8 digits. But here, if we have 0.00021605, that is 8 decimal places. Let me count:

After decimal: 0 (1st), 0 (2nd), 0 (3rd), 2 (4th), 1 (5th), 6 (6th), 0 (7th), 5 (8th). So, 8 digits.

But in the number 0.00021605, the first three zeros after the decimal are positions 1, 2, 3, then 2, 1, 6, 0, 5. Wait, that's 8 digits.

Wait, 0.00021605 is:

- 0.00021605

Breaking it down:

- 0.0 (tenths)
- 0.00 (hundredths)
- 0.000 (thousandths)
- 0.0002 (ten-thousandths)
- 0.00021 (hundred-thousandths)
- 0.000216 (millionths)
- 0.0002160 (ten-millionths)
- 0.00021605 (hundred-millionths)

So, the decimal terminates at the 8th decimal place. Therefore, the decimal representation is 0.00021605, with exactly 8 decimal digits.

Therefore, the digits after the decimal are 0, 0, 0, 2, 1, 6, 0, 5.

But if we consider the entire decimal representation, including the integer part, which is 0, then the digits are 0 (before decimal) and then the decimal digits. However, in the context of the problem, when they say "the sum of the digits in the terminating decimal representation", they might be referring to the digits after the decimal point.

But to be thorough, let me check both possibilities.

Case 1: Sum of digits after decimal: 0 + 0 + 0 + 2 + 1 + 6 + 0 + 5 = 0+0+0+2+1+6+0+5 = 14.

Case 2: Sum of all digits including the integer part: 0 (from integer part) + 0 + 0 + 0 + 2 + 1 + 6 + 0 + 5 = same as above, 14.

Wait, because the integer part is 0, so adding that doesn't change the sum.

But maybe the problem is considering the decimal representation as a string, which would include the leading zero before the decimal? But in reality, the number is 0.00021605, so the digits are 0, 0, 0, 2, 1, 6, 0, 5. However, the leading zero before the decimal is not considered as part of the decimal digits. Wait, actually, in the decimal representation, the digits are both before and after the decimal.

But in the case of 0.00021605, the digits are:

- 0 (units place)
- 0 (tenths)
- 0 (hundredths)
- 0 (thousandths)
- 2 (ten-thousandths)
- 1 (hundred-thousandths)
- 6 (millionths)
- 0 (ten-millionths)
- 5 (hundred-millionths)

Wait, but the decimal terminates after 8 digits, so the units place is 0, and then 8 decimal digits. So, total digits: 1 (units) + 8 (decimal) = 9 digits.

But if we consider all digits, the sum would be 0 (units) + 0 + 0 + 0 + 2 + 1 + 6 + 0 + 5 = 14.

But if the problem is referring only to the decimal part, then it's 0 + 0 + 0 + 2 + 1 + 6 + 0 + 5 = 14.

Either way, the sum is 14.

But let me check again. Wait, the decimal representation is 0.00021605. If we write it as a decimal number, it's 0.00021605. The digits are 0, 0, 0, 2, 1, 6, 0, 5. So, 8 digits after the decimal.

But if you consider the entire number, including the leading zero before the decimal, then it's 0.00021605, which has digits 0, 0, 0, 0, 2, 1, 6, 0, 5. Wait, but that would be 9 digits. Wait, confusion again.

Wait, the number is 0.00021605. Let me write it with all the digits:

- The integer part is 0.
- The decimal part is 00021605.

So, the decimal part has 8 digits: 0, 0, 0, 2, 1, 6, 0, 5.

Therefore, the digits in the decimal representation are the integer part (0) and the decimal part (0, 0, 0, 2, 1, 6, 0, 5).

But if we are to sum all digits, it's 0 + 0 + 0 + 0 + 2 + 1 + 6 + 0 + 5 = 14.

But if the problem is referring only to the decimal part, then it's 0 + 0 + 0 + 2 + 1 + 6 + 0 + 5 = 14.

Either way, the sum is 14.

But let me check once more. Let me think of another example. Suppose the decimal is 0.123. The digits are 0, 1, 2, 3. If we sum them, it's 0 + 1 + 2 + 3 = 6. But if we consider only the decimal part, it's 1 + 2 + 3 = 6.

But if the number is 123.456, the digits are 1, 2, 3, 4, 5, 6. Sum is 21.

So, in our case, the number is 0.00021605. If we consider all digits, including the leading zero before the decimal, then it's 0 (from integer part) and then the decimal digits. But the integer part is 0, so adding that doesn't change the sum.

But in reality, the decimal representation is 0.00021605, so the digits are 0, 0, 0, 2, 1, 6, 0, 5. Wait, but how many digits is that?

If you write the number as 0.00021605, then the digits are:

- 0 (before decimal)
- 0 (tenths)
- 0 (hundredths)
- 0 (thousandths)
- 2 (ten-thousandths)
- 1 (hundred-thousandths)
- 6 (millionths)
- 0 (ten-millionths)
- 5 (hundred-millionths)

So, that's 9 digits. But the decimal terminates after 8 digits. Wait, but the 8th digit is 5. So, the decimal is 0.00021605, which is 8 decimal places.

But the leading zero before the decimal is part of the number, but in terms of digits, it's just one digit. So, the total digits in the decimal representation are 1 (integer part) + 8 (decimal part) = 9 digits.

But if the problem is asking for the sum of the digits in the decimal representation, then it's 0 + 0 + 0 + 0 + 2 + 1 + 6 + 0 + 5 = 14.

But maybe the problem is considering only the decimal part, excluding the integer part. However, in that case, the digits are 0, 0, 0, 2, 1, 6, 0, 5, which sum to 14.

Either way, the answer is 14.

But let me check once again. Maybe I made a mistake in calculating the decimal.

Original fraction: 4321 / (5^7 * 2^8).

We converted it to 21605 / 10^8 = 0.00021605.

But let me verify this with another approach. Let me compute 4321 divided by (5^7 * 2^8).

First, compute denominator: 5^7 is 78125, 2^8 is 256. Multiply them: 78125 * 256.

Let me compute 78125 * 256.

Well, 78125 * 256 = 78125 * (200 + 50 + 6) = 78125*200 + 78125*50 + 78125*6.

Compute each term:

78125 * 200 = 78125 * 2 * 100 = 156250 * 100 = 15,625,000

78125 * 50 = 78125 * 5 * 10 = 390,625 * 10 = 3,906,250

78125 * 6 = 468,750

Now, add them up:

15,625,000 + 3,906,250 = 19,531,250

19,531,250 + 468,750 = 20,000,000

Wait, that's interesting. So, 78125 * 256 = 20,000,000.

So, denominator is 20,000,000.

Therefore, the original fraction is 4321 / 20,000,000.

So, 4321 divided by 20,000,000.

Let me compute that.

Dividing 4321 by 20,000,000.

Well, 4321 / 20,000,000 = (4321 / 20) / 1,000,000.

Compute 4321 / 20:

4321 divided by 20.

Well, 20 * 216 = 4320.

So, 4321 / 20 = 216 + 1/20 = 216.05.

Therefore, 216.05 / 1,000,000 = 0.00021605.

Yes! So, that confirms the decimal is indeed 0.00021605.

Therefore, the decimal representation is 0.00021605.

So, digits after decimal: 0, 0, 0, 2, 1, 6, 0, 5.

Sum: 0 + 0 + 0 + 2 + 1 + 6 + 0 + 5 = 14.

Therefore, the answer is 14.

But let me check once more for errors.

Wait, 0.00021605. Let me write it out:

Position 1: 0

Position 2: 0

Position 3: 0

Position 4: 2

Position 5: 1

Position 6: 6

Position 7: 0

Position 8: 5

So, digits are 0, 0, 0, 2, 1, 6, 0, 5.

Adding them up: 0+0+0+2+1+6+0+5.

Let me compute step by step:

Start with 0.

Add first 0: 0

Add second 0: 0

Add third 0: 0

Add 2: 2

Add 1: 3

Add 6: 9

Add 0: 9

Add 5: 14.

Yes, that's 14.

Therefore, the sum of the digits is 14.

But just to be 100% sure, let me check if there's a possibility that the decimal has more digits or if I made a mistake in the conversion.

Earlier, we saw that 4321 / (5^7 * 2^8) = 4321 / 20,000,000 = 0.00021605.

Alternatively, let me compute 4321 divided by 20,000,000 using another method.

Dividing 4321 by 20,000,000.

Well, 20,000,000 goes into 4321 zero times. So, 0.

Then, 20,000,000 goes into 43,210 (adding a decimal and a zero) 2 times (2*20,000,000 = 40,000,000). Subtract: 43,210 - 40,000,000? Wait, no, this is confusing.

Wait, maybe better to think in terms of decimal places.

Since 20,000,000 is 2 * 10^7.

So, 4321 / (2 * 10^7) = (4321 / 2) / 10^7 = 2160.5 / 10^7 = 0.00021605.

Yes, same result.

Therefore, the decimal is definitely 0.00021605.

Therefore, the digits are as above, sum is 14.

Therefore, the answer is 14.

But just to make sure, let me check if there's a different interpretation.

Suppose someone thought that the decimal is 0.00021605, but maybe there's a trailing zero? For example, if the decimal was 0.000216050, but since it's terminating, it would be 0.00021605.

But no, since we have exactly 8 decimal digits, and the last digit is 5.

Alternatively, maybe I miscalculated 4321 * 5? Let me check again.

4321 * 5:

Well, 4000 * 5 = 20,000

300 * 5 = 1,500

20 * 5 = 100

1 * 5 = 5

Adding them up: 20,000 + 1,500 = 21,500; 21,500 + 100 = 21,600; 21,600 + 5 = 21,605.

Yes, that's correct.

So, 21,605 / 10^8 = 0.00021605.

Therefore, the decimal is correct.

Therefore, the digits are 0, 0, 0, 2, 1, 6, 0, 5.

Sum is 14.

Therefore, I think the answer is 14.

**Final Answer**
\boxed{14}

To find the **sum of the digits** in the **terminating decimal representation** of the fraction:

$$
\frac{4321}{5^7 \cdot 2^8}
$$

we begin by analyzing the structure of the denominator. Since the denominator is composed of only powers of 2 and 5, the decimal representation of the fraction will **terminate**. The number of decimal places is determined by the **maximum exponent** between the powers of 2 and 5 in the denominator.

---

### Step 1: Determine the Number of Decimal Places

The denominator is:

$$
5^7 \cdot 2^8
$$

To convert this into a power of 10, we need to balance the powers of 2 and 5. Since the exponent of 2 is greater (8), we multiply numerator and denominator by $5^1$ to make the denominator $10^8$:

$$
\frac{4321}{5^7 \cdot 2^8} = \frac{4321 \cdot 5}{5^8 \cdot 2^8} = \frac{21605}{10^8}
$$

This means the decimal representation will have **8 decimal places**.

---

### Step 2: Convert the Fraction to Decimal

$$
\frac{21605}{10^8} = 0.00021605
$$

This is the **terminating decimal representation** of the fraction.

---

### Step 3: Identify the Digits

The decimal is:

$$
0.00021605
$$

The digits after the decimal point are:

$$
0, 0, 0, 2, 1, 6, 0, 5
$$

---

### Step 4: Compute the Sum of the Digits

Add the digits:

$$
0 + 0 + 0 + 2 + 1 + 6 + 0 + 5 = 14
$$

---

### Final Answer

$$
\boxed{14}
$$

[Assistant]
Explanation:
Okay, so I need to find the sum of the digits in the terminating decimal representation of the fraction 4321 divided by (5^7 times 2^8). Hmm, let me think about how to approach this.

First, I remember that when you have a fraction with denominators that are powers of 2 and/or 5, the decimal will terminate. That's because 2 and 5 are the prime factors of 10, the base of our number system. So, the decimal will terminate, and the number of decimal places will be determined by the maximum exponent between the powers of 2 and 5 in the denominator.

Wait, let me recall the exact rule. If you have a fraction a/b where b is of the form 2^n * 5^m, then the decimal terminates after max(n, m) digits. So, in this case, the denominator is 5^7 * 2^8. Therefore, the maximum of 7 and 8 is 8. So, the decimal will terminate after 8 digits. That means if I convert this fraction to a decimal, it will have 8 digits after the decimal point, and then it will end.

But how do I actually compute the decimal? Well, one way is to multiply numerator and denominator by something to make the denominator a power of 10. Since the denominator is 5^7 * 2^8, if I multiply numerator and denominator by 5^(something) to make the denominator 10^8? Wait, let me check.

The denominator is 5^7 * 2^8. If I want to make it 10^k, then I need to have both 2^k and 5^k. Since the denominator already has 2^8 and 5^7, to make it 10^8, I need to multiply by 5^(8 - 7) = 5^1. Because 5^7 * 5^1 = 5^8, and 2^8 * 5^1 = 2^8 * 5^1. Wait, no, that would make the denominator 2^8 * 5^8, which is 10^8. So, if I multiply numerator and denominator by 5^1, then the denominator becomes 10^8, and the numerator becomes 4321 * 5.

Let me write that down:

Original fraction: 4321 / (5^7 * 2^8)

Multiply numerator and denominator by 5^1:

Numerator: 4321 * 5

Denominator: 5^7 * 2^8 * 5^1 = 5^(7+1) * 2^8 = 5^8 * 2^8 = (5*2)^8 = 10^8

So, the fraction becomes (4321 * 5) / 10^8. Therefore, the decimal representation is (4321 * 5) divided by 10^8, which is the same as moving the decimal point 8 places to the left in the numerator.

So, first, let me compute 4321 * 5. Let me do that multiplication:

4321 * 5:

Well, 4000 * 5 = 20,000

300 * 5 = 1,500

20 * 5 = 100

1 * 5 = 5

Adding them up: 20,000 + 1,500 = 21,500; 21,500 + 100 = 21,600; 21,600 + 5 = 21,605.

So, 4321 * 5 = 21,605.

Therefore, the fraction is 21,605 / 10^8.

So, 21,605 divided by 10^8 is 0.000021605. Wait, let me check that.

Dividing by 10^8 moves the decimal point 8 places to the left. Let me write 21,605 as 21605.0.

Starting from the right, moving decimal 8 places left:

Original number: 21605.0

Move decimal 1 place left: 2160.50

2 places: 216.050

3 places: 21.6050

4 places: 2.16050

5 places: 0.216050

6 places: 0.0216050

7 places: 0.00216050

8 places: 0.000216050

Wait, so that would be 0.00021605? But wait, the numerator is 21,605. So, 21,605 divided by 10^8 is 0.00021605? Let me check with another approach.

Alternatively, 10^8 is 100,000,000. So, 21,605 divided by 100,000,000.

So, 21,605 / 100,000,000 = 0.00021605.

Yes, because 21,605 is 2.1605 x 10^4, so divided by 10^8 is 2.1605 x 10^-4, which is 0.00021605.

But wait, let me count the decimal places. Let me write 21,605 as 21605. So, to write it as a decimal with 8 decimal places, we need to add leading zeros.

So, 21605 is the same as 000021605. If we have 8 decimal places, then it's 0.000021605? Wait, hold on. Wait, confusion here.

Wait, if I have 21,605 divided by 10^8, then it's equal to 21,605 divided by 100,000,000. Let me write that as a decimal.

Let me think of 21,605 divided by 100,000,000.

Dividing by 10^8 is moving the decimal point 8 places to the left.

Starting with 21,605.0

First decimal place: 2160.50

Second: 216.050

Third: 21.6050

Fourth: 2.16050

Fifth: 0.216050

Sixth: 0.0216050

Seventh: 0.00216050

Eighth: 0.000216050

So, after moving the decimal 8 places, it's 0.00021605. But wait, the original number is 21,605. So, how many digits do we have? Let me think.

The number 21,605 has 5 digits. When we divide by 10^8, we need to add 8 - 5 = 3 leading zeros? Wait, no. Wait, if you have a number with n digits, dividing by 10^k will result in a decimal with k digits after the decimal point, but depending on the number.

Alternatively, think of 21,605 as 21605. So, to write this as a decimal with 8 decimal places, we need to add 3 zeros before the decimal point? Wait, no. Let me think of it as:

The decimal representation is 0. followed by (number of zeros) then the digits of the numerator.

Since 21,605 is 21605, which is 5 digits. To have 8 decimal places, we need to have 8 digits after the decimal. So, starting from the right, we need to add 3 zeros before the digits.

Wait, for example, if I have 123 divided by 10^5, that's 0.00123. So, 123 is 3 digits, divided by 10^5, so 5 decimal places: 0.00123. So, adding two zeros before the 123.

Similarly, 21,605 divided by 10^8 is 0.00021605. Let me count the decimal places:

0.00021605 has 8 decimal places. Let me count:

First decimal place: 0

Second: 0

Third: 0

Fourth: 2

Fifth: 1

Sixth: 6

Seventh: 0

Eighth: 5

Wait, but that's only 8 digits. Wait, but 21,605 is 5 digits. So, how does that work?

Wait, 21,605 divided by 10^8 is equal to 0.00021605. Let me check with actual division.

Let me write 21,605 divided by 100,000,000.

So, 21,605 ÷ 100,000,000.

Well, 100,000,000 goes into 21,605 zero times. So, 0. Then, we add decimal places.

How many decimal places? Let me think.

Alternatively, think of 21,605 as 2.1605 x 10^4. Then, dividing by 10^8 is 2.1605 x 10^(4 - 8) = 2.1605 x 10^-4 = 0.00021605.

Yes, that's correct. So, 0.00021605.

So, the decimal is 0.00021605.

But wait, let me check if there are more digits. Because 21,605 is 5 digits, and we have 8 decimal places. So, the decimal would be 0.00021605. Let me count the digits after the decimal: 0, 0, 0, 2, 1, 6, 0, 5. That's 8 digits.

So, the decimal is 0.00021605.

But wait, is that correct? Let me check with another method.

Alternatively, since the denominator is 5^7 * 2^8, which is 5^7 * 2^7 * 2^1 = (5*2)^7 * 2 = 10^7 * 2. Therefore, the denominator is 10^7 * 2.

So, the original fraction is 4321 / (10^7 * 2) = (4321 / 2) / 10^7.

So, 4321 divided by 2 is 2160.5. Then, divided by 10^7 is 0.00021605.

Yes! That's the same result. So, 0.00021605.

So, the decimal is 0.00021605.

Now, the question is asking for the sum of the digits in the terminating decimal representation.

So, first, let me write out the decimal: 0.00021605.

But wait, is that all? Let me check if there are more digits. Since the decimal terminates after 8 digits, as we determined earlier.

But let me confirm. The decimal is 0.00021605. Let me count the digits after the decimal:

Position 1: 0

Position 2: 0

Position 3: 0

Position 4: 2

Position 5: 1

Position 6: 6

Position 7: 0

Position 8: 5

So, that's 8 digits. So, the decimal is 0.00021605.

But wait, when I did the calculation earlier, I thought of it as 0.00021605. However, when I first thought of moving the decimal, I thought of 0.00021605. But let me check if there's a trailing zero or something.

Wait, 21,605 divided by 10^8 is 0.00021605. So, that's exactly 8 decimal digits.

But let me check with another approach. Let me compute 4321 divided by (5^7 * 2^8).

Alternatively, compute denominator first: 5^7 is 78125, and 2^8 is 256. So, 78125 * 256. Let me compute that.

But that might be tedious, but maybe I can do it step by step.

First, 78125 * 256. Let me note that 256 is 2^8, which is 256.

Alternatively, 78125 * 256 = 78125 * (200 + 50 + 6) = 78125*200 + 78125*50 + 78125*6.

But maybe there's a better way.

Alternatively, note that 78125 * 256 = 78125 * 2^8. Since 78125 is 5^7, so 5^7 * 2^8 = 10^7 * 2.

Wait, that's the same as before. So, denominator is 10^7 * 2.

So, 4321 divided by (10^7 * 2) is equal to (4321 / 2) / 10^7 = 2160.5 / 10^7 = 0.00021605.

Yes, same result.

So, decimal is 0.00021605.

Therefore, the digits after the decimal are 0, 0, 0, 2, 1, 6, 0, 5.

So, the digits are: 0, 0, 0, 2, 1, 6, 0, 5.

Now, the question says "the sum of the digits in the terminating decimal representation".

But wait, does that include the digits before the decimal? Well, the decimal representation is 0.00021605. The digits before the decimal are just 0, which is a single digit. But usually, when they say "sum of the digits in the decimal representation", they might be referring to the digits after the decimal point. However, sometimes they might include the integer part.

But let me check the problem statement again: "the sum of the digits in the terminating decimal representation of the fraction".

The decimal representation is 0.00021605. So, the digits are 0 (before decimal), then 0, 0, 0, 2, 1, 6, 0, 5.

But if we consider the entire decimal representation, including the integer part, then the digits are 0, 0, 0, 2, 1, 6, 0, 5. However, the integer part is 0, so adding that would be 0 + 0 + 0 + 2 + 1 + 6 + 0 + 5.

But maybe the problem is referring only to the digits after the decimal? Let me think.

In some contexts, when they say "decimal representation", they might mean the fractional part. But to be safe, let me check both possibilities.

But let me think again. The fraction is 4321/(5^7*2^8). When converted to decimal, it's 0.00021605. So, the entire decimal representation is 0.00021605. The digits are 0, 0, 0, 2, 1, 6, 0, 5.

But if we consider the entire number, including the leading zero before the decimal, then the digits are 0 (before decimal) and then the decimal digits. However, in decimal representations, leading zeros before the decimal are not considered as part of the digits in the number. For example, the number 0.00021605 is just 0.00021605, and the digits are 0, 0, 0, 2, 1, 6, 0, 5.

But if you consider the entire decimal expansion, including the integer part, then the digits are 0 (from the integer part) and then the decimal digits. However, the integer part is 0, so adding that would be 0 + 0 + 0 + 2 + 1 + 6 + 0 + 5.

But maybe the problem is referring only to the digits after the decimal point. Let me check the problem statement again: "the sum of the digits in the terminating decimal representation".

The term "decimal representation" usually includes all digits, both before and after the decimal. However, in some contexts, especially when talking about decimal expansions, people might refer to the fractional part.

But to be precise, let me check with an example. For instance, if the decimal was 0.123, the digits are 0, 1, 2, 3. But if you consider the entire number, it's 0.123, so digits are 0, 1, 2, 3. However, if you have a number like 123.456, the digits are 1, 2, 3, 4, 5, 6.

But in our case, the number is 0.00021605. So, the digits are 0, 0, 0, 2, 1, 6, 0, 5. However, the leading zero before the decimal is part of the decimal representation. But in reality, the number is 0.00021605, so the digits are 0 (before decimal), then 0, 0, 0, 2, 1, 6, 0, 5.

But if you consider the decimal representation as a string, it's "0.00021605", so the digits are 0, 0, 0, 2, 1, 6, 0, 5. However, the leading zero before the decimal is not considered as part of the decimal digits. Wait, actually, in the decimal representation, the digits are both before and after the decimal.

But in the case of 0.00021605, the digits are 0 (units place), 0 (tenths), 0 (hundredths), 0 (thousandths), 2 (ten-thousandths), 1 (hundred-thousandths), 6 (millionths), 0 (ten-millionths), 5 (hundred-millionths).

But if you consider the entire decimal representation, the digits are 0, 0, 0, 0, 2, 1, 6, 0, 5. Wait, but that would be 9 digits. Wait, confusion arises here.

Wait, the number is 0.00021605. Let me write it out with all decimal places:

Position: 1: 0 (tenths)

Position 2: 0 (hundredths)

Position 3: 0 (thousandths)

Position 4: 0 (ten-thousandths)

Position 5: 2 (hundred-thousandths)

Position 6: 1 (millionths)

Position 7: 6 (ten-millionths)

Position 8: 0 (hundred-millionths)

Position 9: 5 (billionths)

Wait, but the decimal terminates after 8 digits. Wait, earlier we said that the decimal terminates after 8 digits. But here, if we have 0.00021605, that is 8 decimal places. Let me count:

After decimal: 0 (1st), 0 (2nd), 0 (3rd), 2 (4th), 1 (5th), 6 (6th), 0 (7th), 5 (8th). So, 8 digits.

But in the number 0.00021605, the first three zeros after the decimal are positions 1, 2, 3, then 2, 1, 6, 0, 5. Wait, that's 8 digits.

Wait, 0.00021605 is:

- 0.00021605

Breaking it down:

- 0.0 (tenths)
- 0.00 (hundredths)
- 0.000 (thousandths)
- 0.0002 (ten-thousandths)
- 0.00021 (hundred-thousandths)
- 0.000216 (millionths)
- 0.0002160 (ten-millionths)
- 0.00021605 (hundred-millionths)

So, the decimal terminates at the 8th decimal place. Therefore, the decimal representation is 0.00021605, with exactly 8 decimal digits.

Therefore, the digits after the decimal are 0, 0, 0, 2, 1, 6, 0, 5.

But if we consider the entire decimal representation, including the integer part, which is 0, then the digits are 0 (before decimal) and then the decimal digits. However, in the context of the problem, when they say "the sum of the digits in the terminating decimal representation", they might be referring to the digits after the decimal point.

But to be thorough, let me check both possibilities.

Case 1: Sum of digits after decimal: 0 + 0 + 0 + 2 + 1 + 6 + 0 + 5 = 0+0+0+2+1+6+0+5 = 14.

Case 2: Sum of all digits including the integer part: 0 (from integer part) + 0 + 0 + 0 + 2 + 1 + 6 + 0 + 5 = same as above, 14.

Wait, because the integer part is 0, so adding that doesn't change the sum.

But maybe the problem is considering the decimal representation as a string, which would include the leading zero before the decimal? But in reality, the number is 0.00021605, so the digits are 0, 0, 0, 2, 1, 6, 0, 5. However, the leading zero before the decimal is not considered as part of the decimal digits. Wait, actually, in the decimal representation, the digits are both before and after the decimal.

But in the case of 0.00021605, the digits are:

- 0 (units place)
- 0 (tenths)
- 0 (hundredths)
- 0 (thousandths)
- 2 (ten-thousandths)
- 1 (hundred-thousandths)
- 6 (millionths)
- 0 (ten-millionths)
- 5 (hundred-millionths)

Wait, but the decimal terminates after 8 digits, so the units place is 0, and then 8 decimal digits. So, total digits: 1 (units) + 8 (decimal) = 9 digits.

But if we consider all digits, the sum would be 0 (units) + 0 + 0 + 0 + 2 + 1 + 6 + 0 + 5 = 14.

But if the problem is referring only to the decimal part, then it's 0 + 0 + 0 + 2 + 1 + 6 + 0 + 5 = 14.

Either way, the sum is 14.

But let me check again. Wait, the decimal representation is 0.00021605. If we write it as a decimal number, it's 0.00021605. The digits are 0, 0, 0, 2, 1, 6, 0, 5. So, 8 digits after the decimal.

But if you consider the entire number, including the leading zero before the decimal, then it's 0.00021605, which has digits 0, 0, 0, 0, 2, 1, 6, 0, 5. Wait, but that would be 9 digits. Wait, confusion again.

Wait, the number is 0.00021605. Let me write it with all the digits:

- The integer part is 0.
- The decimal part is 00021605.

So, the decimal part has 8 digits: 0, 0, 0, 2, 1, 6, 0, 5.

Therefore, the digits in the decimal representation are the integer part (0) and the decimal part (0, 0, 0, 2, 1, 6, 0, 5).

But if we are to sum all digits, it's 0 + 0 + 0 + 0 + 2 + 1 + 6 + 0 + 5 = 14.

But if the problem is referring only to the decimal part, then it's 0 + 0 + 0 + 2 + 1 + 6 + 0 + 5 = 14.

Either way, the sum is 14.

But let me check once more. Let me think of another example. Suppose the decimal is 0.123. The digits are 0, 1, 2, 3. If we sum them, it's 0 + 1 + 2 + 3 = 6. But if we consider only the decimal part, it's 1 + 2 + 3 = 6.

But if the number is 123.456, the digits are 1, 2, 3, 4, 5, 6. Sum is 21.

So, in our case, the number is 0.00021605. If we consider all digits, including the leading zero before the decimal, then it's 0 (from integer part) and then the decimal digits. But the integer part is 0, so adding that doesn't change the sum.

But in reality, the decimal representation is 0.00021605, so the digits are 0, 0, 0, 2, 1, 6, 0, 5. Wait, but how many digits is that?

If you write the number as 0.00021605, then the digits are:

- 0 (before decimal)
- 0 (tenths)
- 0 (hundredths)
- 0 (thousandths)
- 2 (ten-thousandths)
- 1 (hundred-thousandths)
- 6 (millionths)
- 0 (ten-millionths)
- 5 (hundred-millionths)

So, that's 9 digits. But the decimal terminates after 8 digits. Wait, but the 8th digit is 5. So, the decimal is 0.00021605, which is 8 decimal places.

But the leading zero before the decimal is part of the number, but in terms of digits, it's just one digit. So, the total digits in the decimal representation are 1 (integer part) + 8 (decimal part) = 9 digits.

But if the problem is asking for the sum of the digits in the decimal representation, then it's 0 + 0 + 0 + 0 + 2 + 1 + 6 + 0 + 5 = 14.

But maybe the problem is considering only the decimal part, excluding the integer part. However, in that case, the digits are 0, 0, 0, 2, 1, 6, 0, 5, which sum to 14.

Either way, the answer is 14.

But let me check once again. Maybe I made a mistake in calculating the decimal.

Original fraction: 4321 / (5^7 * 2^8).

We converted it to 21605 / 10^8 = 0.00021605.

But let me verify this with another approach. Let me compute 4321 divided by (5^7 * 2^8).

First, compute denominator: 5^7 is 78125, 2^8 is 256. Multiply them: 78125 * 256.

Let me compute 78125 * 256.

Well, 78125 * 256 = 78125 * (200 + 50 + 6) = 78125*200 + 78125*50 + 78125*6.

Compute each term:

78125 * 200 = 78125 * 2 * 100 = 156250 * 100 = 15,625,000

78125 * 50 = 78125 * 5 * 10 = 390,625 * 10 = 3,906,250

78125 * 6 = 468,750

Now, add them up:

15,625,000 + 3,906,250 = 19,531,250

19,531,250 + 468,750 = 20,000,000

Wait, that's interesting. So, 78125 * 256 = 20,000,000.

So, denominator is 20,000,000.

Therefore, the original fraction is 4321 / 20,000,000.

So, 4321 divided by 20,000,000.

Let me compute that.

Dividing 4321 by 20,000,000.

Well, 4321 / 20,000,000 = (4321 / 20) / 1,000,000.

Compute 4321 / 20:

4321 divided by 20.

Well, 20 * 216 = 4320.

So, 4321 / 20 = 216 + 1/20 = 216.05.

Therefore, 216.05 / 1,000,000 = 0.00021605.

Yes! So, that confirms the decimal is indeed 0.00021605.

Therefore, the decimal representation is 0.00021605.

So, digits after decimal: 0, 0, 0, 2, 1, 6, 0, 5.

Sum: 0 + 0 + 0 + 2 + 1 + 6 + 0 + 5 = 14.

Therefore, the answer is 14.

But let me check once more for errors.

Wait, 0.00021605. Let me write it out:

Position 1: 0

Position 2: 0

Position 3: 0

Position 4: 2

Position 5: 1

Position 6: 6

Position 7: 0

Position 8: 5

So, digits are 0, 0, 0, 2, 1, 6, 0, 5.

Adding them up: 0+0+0+2+1+6+0+5.

Let me compute step by step:

Start with 0.

Add first 0: 0

Add second 0: 0

Add third 0: 0

Add 2: 2

Add 1: 3

Add 6: 9

Add 0: 9

Add 5: 14.

Yes, that's 14.

Therefore, the sum of the digits is 14.

But just to be 100% sure, let me check if there's a possibility that the decimal has more digits or if I made a mistake in the conversion.

Earlier, we saw that 4321 / (5^7 * 2^8) = 4321 / 20,000,000 = 0.00021605.

Alternatively, let me compute 4321 divided by 20,000,000 using another method.

Dividing 4321 by 20,000,000.

Well, 20,000,000 goes into 4321 zero times. So, 0.

Then, 20,000,000 goes into 43,210 (adding a decimal and a zero) 2 times (2*20,000,000 = 40,000,000). Subtract: 43,210 - 40,000,000? Wait, no, this is confusing.

Wait, maybe better to think in terms of decimal places.

Since 20,000,000 is 2 * 10^7.

So, 4321 / (2 * 10^7) = (4321 / 2) / 10^7 = 2160.5 / 10^7 = 0.00021605.

Yes, same result.

Therefore, the decimal is definitely 0.00021605.

Therefore, the digits are as above, sum is 14.

Therefore, the answer is 14.

But just to make sure, let me check if there's a different interpretation.

Suppose someone thought that the decimal is 0.00021605, but maybe there's a trailing zero? For example, if the decimal was 0.000216050, but since it's terminating, it would be 0.00021605.

But no, since we have exactly 8 decimal digits, and the last digit is 5.

Alternatively, maybe I miscalculated 4321 * 5? Let me check again.

4321 * 5:

Well, 4000 * 5 = 20,000

300 * 5 = 1,500

20 * 5 = 100

1 * 5 = 5

Adding them up: 20,000 + 1,500 = 21,500; 21,500 + 100 = 21,600; 21,600 + 5 = 21,605.

Yes, that's correct.

So, 21,605 / 10^8 = 0.00021605.

Therefore, the decimal is correct.

Therefore, the digits are 0, 0, 0, 2, 1, 6, 0, 5.

Sum is 14.

Therefore, I think the answer is 14.

**Final Answer**
\boxed{14}

To find the **sum of the digits** in the **terminating decimal representation** of the fraction:

$$
\frac{4321}{5^7 \cdot 2^8}
$$

we begin by analyzing the structure of the denominator. Since the denominator is composed of only powers of 2 and 5, the decimal representation of the fraction will **terminate**. The number of decimal places is determined by the **maximum exponent** between the powers of 2 and 5 in the denominator.

---

### Step 1: Determine the Number of Decimal Places

The denominator is:

$$
5^7 \cdot 2^8
$$

To convert this into a power of 10, we need to balance the powers of 2 and 5. Since the exponent of 2 is greater (8), we multiply numerator and denominator by $5^1$ to make the denominator $10^8$:

$$
\frac{4321}{5^7 \cdot 2^8} = \frac{4321 \cdot 5}{5^8 \cdot 2^8} = \frac{21605}{10^8}
$$

This means the decimal representation will have **8 decimal places**.

---

### Step 2: Convert the Fraction to Decimal

$$
\frac{21605}{10^8} = 0.00021605
$$

This is the **terminating decimal representation** of the fraction.

---

### Step 3: Identify the Digits

The decimal is:

$$
0.00021605
$$

The digits after the decimal point are:

$$
0, 0, 0, 2, 1, 6, 0, 5
$$

---

### Step 4: Compute the Sum of the Digits

Add the digits:

$$
0 + 0 + 0 + 2 + 1 + 6 + 0 + 5 = 14
$$

---

### Final Answer

$$
\boxed{14}
$$

Solution: Rewriting $\frac{4321}{5^7\cdot2^8}$ as a decimal with a denominator of $5^8\cdot2^8=10^8$, we have \[ \frac{4321}{5^7\cdot2^8}\cdot\frac{5^{1}}{5^{1}}=\frac{4321\cdot5}{10^8}=\frac{21605}{10^{8}}=0.00021605.\]So, the sum of the digits of the decimal representation is $2+1+6+0+5 = \boxed{14}$.


Explanation:
Okay, so I need to find the value of this expression when x is 4. The expression is (3x - 2)(4x + 1) minus (3x - 2) times 4x plus 1. Let me write that down again to make sure I have it right:

(3x - 2)(4x + 1) - (3x - 2)4x + 1

Wait, hold on. Is that the correct expression? Let me check. The original problem says:

(3x-2)(4x+1)-(3x-2)4x+1

So, it's (3x - 2)(4x + 1) minus (3x - 2) times 4x plus 1. Hmm. So, the way it's written, there's a minus sign before the second term, which is (3x - 2)4x, and then a plus 1 at the end. So, the entire expression is:

[(3x - 2)(4x + 1)] - [(3x - 2) * 4x] + 1

Alternatively, maybe there's a different grouping? Let me check again. The original problem is written as:

(3x-2)(4x+1)-(3x-2)4x+1

So, without parentheses around the entire second part, it's possible that the expression is:

(3x - 2)(4x + 1) - (3x - 2) * 4x + 1

Which would be the same as:

First term: (3x - 2)(4x + 1)

Second term: minus (3x - 2) * 4x

Third term: plus 1

So, that's how I should interpret it. So, maybe I can simplify this expression before plugging in x = 4? Let me try that.

First, let me expand the first term: (3x - 2)(4x + 1). Let me do that step by step.

Multiply 3x by 4x: 3x * 4x = 12x²

Then 3x * 1 = 3x

Then -2 * 4x = -8x

Then -2 * 1 = -2

So, combining those terms: 12x² + 3x - 8x - 2 = 12x² - 5x - 2

Okay, so the first part simplifies to 12x² - 5x - 2.

Now, the second term is -(3x - 2) * 4x. Let me compute that.

First, compute (3x - 2) * 4x. Let me do that:

Multiply 3x by 4x: 12x²

Then -2 * 4x: -8x

So, that gives 12x² - 8x. Then, since there's a minus sign in front of that term, it becomes -12x² + 8x.

So, putting that together with the first part:

First part: 12x² - 5x - 2

Second part: -12x² + 8x

Third term: +1

So, combining all three parts:

(12x² - 5x - 2) + (-12x² + 8x) + 1

Let me combine like terms.

First, 12x² - 12x² = 0. So those cancel out.

Then, -5x + 8x = 3x

Then, -2 + 1 = -1

So, overall, the entire expression simplifies to 3x - 1.

Wait, that seems too simple. Let me check my steps again to make sure I didn't make a mistake.

Original expression:

(3x - 2)(4x + 1) - (3x - 2)4x + 1

First term expansion: 12x² -5x -2. Correct.

Second term: -(3x - 2)*4x. Let me check that again. (3x - 2)*4x is 12x² - 8x. Then with the negative sign, it's -12x² + 8x. Correct.

Third term is +1. So combining:

First term: 12x² -5x -2

Second term: -12x² +8x

Third term: +1

Adding them together:

12x² -12x² = 0

-5x +8x = 3x

-2 +1 = -1

So total expression simplifies to 3x -1. So, if that's the case, then when x=4, it's 3*4 -1 = 12 -1 = 11. So the answer is 11?

But let me verify this by plugging x=4 directly into the original expression, just to make sure that my simplification is correct.

Original expression:

(3x - 2)(4x + 1) - (3x - 2)4x + 1

Let me compute each part when x=4.

First, compute (3x - 2):

3*4 -2 = 12 -2 = 10

Then compute (4x +1):

4*4 +1 = 16 +1 = 17

So first term is 10 * 17 = 170

Second term: -(3x -2)*4x. Let's compute (3x -2)*4x:

We already know 3x -2 is 10, and 4x is 16. So 10 * 16 = 160. Then with the negative sign, it's -160.

Third term is +1.

So total expression is 170 - 160 +1 = 170 -160 is 10, plus 1 is 11. So that's 11. Which matches the simplified expression 3x -1 when x=4: 3*4 -1=12-1=11. So that seems correct.

But let me check again if my simplification was correct. Because sometimes when you factor or simplify, you might make a mistake.

Original expression:

(3x - 2)(4x + 1) - (3x - 2)4x + 1

Alternatively, maybe I can factor out (3x - 2) from the first two terms?

Let me try that. Let's see:

First two terms: (3x - 2)(4x + 1) - (3x - 2)4x

Factor out (3x - 2):

(3x - 2)[(4x +1) - 4x] +1

Compute inside the brackets: (4x +1 -4x) = 1

So that becomes (3x -2)(1) +1 = 3x -2 +1 = 3x -1. Which is the same as before. So that's another way to see it. So that confirms that the expression simplifies to 3x -1. Therefore, when x=4, it's 11. So that's the answer.

But just to be thorough, let me compute the original expression step by step without simplifying first. Let me do that again with x=4.

Original expression:

(3x - 2)(4x + 1) - (3x - 2)4x + 1

Compute each part:

First part: (3x - 2)(4x + 1). As before, with x=4, that's 10 * 17 = 170.

Second part: -(3x - 2) * 4x. Which is -10 * 16 = -160.

Third part: +1.

So total is 170 -160 +1 = 11. Yep, same result.

Alternatively, if I didn't simplify, but just compute each term:

Alternatively, maybe I can compute the entire expression step by step:

First, compute (3x - 2)(4x + 1):

Let me compute that again with x=4:

(3*4 -2) = 10, (4*4 +1)=17, 10*17=170.

Then subtract (3x -2)*4x: which is 10*16=160. So 170 - 160 = 10.

Then add 1: 10 +1=11. Yep, same answer.

So regardless of the method, the answer is 11. Therefore, I think that's correct.

But just to make sure, let me try plugging x=4 into the simplified expression 3x -1:

3*4 -1 = 12 -1 = 11. Correct.

So, all methods lead to 11. Therefore, the answer is 11.

But just to check if there was any possible misinterpretation of the original expression. For example, if the original expression was meant to be [(3x -2)(4x +1) - (3x -2)4x] +1, which is how I interpreted it, but if someone misreads it as (3x -2)(4x +1 - (3x -2)4x) +1, that would be different. But in that case, the parentheses would be different. Let me check.

Original problem: (3x-2)(4x+1)-(3x-2)4x+1

If someone misreads it as (3x -2)(4x +1 - (3x -2)4x) +1, that would be different. Let me check if that's possible.

But according to order of operations, multiplication is done before addition and subtraction, unless parentheses dictate otherwise. So, the original expression is:

First term: (3x -2)(4x +1)

Second term: - (3x -2) * 4x

Third term: +1

So, the expression is:

First term - second term + third term.

So, unless there are parentheses around the entire second term and third term, which there aren't, so the original expression is as I interpreted.

But just to check, if someone thought that the entire expression is (3x -2)(4x +1 - (3x -2)4x) +1, then that would be different. Let me compute that for x=4, just in case.

Compute inside the parentheses: 4x +1 - (3x -2)4x.

With x=4:

4*4 +1 - (3*4 -2)*4*4 = 16 +1 - (10)*16 = 17 - 160 = -143

Then multiply by (3x -2) which is 10: 10*(-143) = -1430

Then add 1: -1430 +1 = -1429. Which is way different. But that's not the case here, because the original expression doesn't have parentheses around the 4x +1 - (3x -2)4x. So, that's not the correct interpretation.

Therefore, my original interpretation is correct, and the answer is 11.

Alternatively, maybe the original problem had different parentheses? But as written, it's (3x-2)(4x+1)-(3x-2)4x+1. So, no, the way it's written, the expression is as I have.

Therefore, I think the answer is 11. So, I can confidently say that the value is 11 when x=4.

**Final Answer**
The value of the expression when $ x = 4 $ is \boxed{11}.

To evaluate the expression:

$$
(3x - 2)(4x + 1) - (3x - 2) \cdot 4x + 1
$$

when $ x = 4 $, we can either simplify the expression first or substitute $ x = 4 $ directly. Let's proceed step by step.

---

### **Step 1: Simplify the Expression**

We begin by simplifying the expression algebraically.

#### **First Term:**
$$
(3x - 2)(4x + 1)
$$

Use the distributive property (FOIL method):

$$
= 3x \cdot 4x + 3x \cdot 1 - 2 \cdot 4x - 2 \cdot 1
= 12x^2 + 3x - 8x - 2
= 12x^2 - 5x - 2
$$

#### **Second Term:**
$$
-(3x - 2) \cdot 4x
$$

First compute the product:

$$
(3x - 2) \cdot 4x = 12x^2 - 8x
$$

Now apply the negative sign:

$$
- (12x^2 - 8x) = -12x^2 + 8x
$$

#### **Third Term:**
$$
+1
$$

---

### **Step 2: Combine All Terms**

Now combine all the simplified parts:

$$
(12x^2 - 5x - 2) + (-12x^2 + 8x) + 1
$$

Group like terms:

- $ 12x^2 - 12x^2 = 0 $
- $ -5x + 8x = 3x $
- $ -2 + 1 = -1 $

So the entire expression simplifies to:

$$
3x - 1
$$

---

### **Step 3: Substitute $ x = 4 $**

Now plug in $ x = 4 $ into the simplified expression:

$$
3(4) - 1 = 12 - 1 = 11
$$

---

### **Final Answer**

$$
\boxed{11}
$$

[Assistant]
Explanation:
Okay, so I need to find the value of this expression when x is 4. The expression is (3x - 2)(4x + 1) minus (3x - 2) times 4x plus 1. Let me write that down again to make sure I have it right:

(3x - 2)(4x + 1) - (3x - 2)4x + 1

Wait, hold on. Is that the correct expression? Let me check. The original problem says:

(3x-2)(4x+1)-(3x-2)4x+1

So, it's (3x - 2)(4x + 1) minus (3x - 2) times 4x plus 1. Hmm. So, the way it's written, there's a minus sign before the second term, which is (3x - 2)4x, and then a plus 1 at the end. So, the entire expression is:

[(3x - 2)(4x + 1)] - [(3x - 2) * 4x] + 1

Alternatively, maybe there's a different grouping? Let me check again. The original problem is written as:

(3x-2)(4x+1)-(3x-2)4x+1

So, without parentheses around the entire second part, it's possible that the expression is:

(3x - 2)(4x + 1) - (3x - 2) * 4x + 1

Which would be the same as:

First term: (3x - 2)(4x + 1)

Second term: minus (3x - 2) * 4x

Third term: plus 1

So, that's how I should interpret it. So, maybe I can simplify this expression before plugging in x = 4? Let me try that.

First, let me expand the first term: (3x - 2)(4x + 1). Let me do that step by step.

Multiply 3x by 4x: 3x * 4x = 12x²

Then 3x * 1 = 3x

Then -2 * 4x = -8x

Then -2 * 1 = -2

So, combining those terms: 12x² + 3x - 8x - 2 = 12x² - 5x - 2

Okay, so the first part simplifies to 12x² - 5x - 2.

Now, the second term is -(3x - 2) * 4x. Let me compute that.

First, compute (3x - 2) * 4x. Let me do that:

Multiply 3x by 4x: 12x²

Then -2 * 4x: -8x

So, that gives 12x² - 8x. Then, since there's a minus sign in front of that term, it becomes -12x² + 8x.

So, putting that together with the first part:

First part: 12x² - 5x - 2

Second part: -12x² + 8x

Third term: +1

So, combining all three parts:

(12x² - 5x - 2) + (-12x² + 8x) + 1

Let me combine like terms.

First, 12x² - 12x² = 0. So those cancel out.

Then, -5x + 8x = 3x

Then, -2 + 1 = -1

So, overall, the entire expression simplifies to 3x - 1.

Wait, that seems too simple. Let me check my steps again to make sure I didn't make a mistake.

Original expression:

(3x - 2)(4x + 1) - (3x - 2)4x + 1

First term expansion: 12x² -5x -2. Correct.

Second term: -(3x - 2)*4x. Let me check that again. (3x - 2)*4x is 12x² - 8x. Then with the negative sign, it's -12x² + 8x. Correct.

Third term is +1. So combining:

First term: 12x² -5x -2

Second term: -12x² +8x

Third term: +1

Adding them together:

12x² -12x² = 0

-5x +8x = 3x

-2 +1 = -1

So total expression simplifies to 3x -1. So, if that's the case, then when x=4, it's 3*4 -1 = 12 -1 = 11. So the answer is 11?

But let me verify this by plugging x=4 directly into the original expression, just to make sure that my simplification is correct.

Original expression:

(3x - 2)(4x + 1) - (3x - 2)4x + 1

Let me compute each part when x=4.

First, compute (3x - 2):

3*4 -2 = 12 -2 = 10

Then compute (4x +1):

4*4 +1 = 16 +1 = 17

So first term is 10 * 17 = 170

Second term: -(3x -2)*4x. Let's compute (3x -2)*4x:

We already know 3x -2 is 10, and 4x is 16. So 10 * 16 = 160. Then with the negative sign, it's -160.

Third term is +1.

So total expression is 170 - 160 +1 = 170 -160 is 10, plus 1 is 11. So that's 11. Which matches the simplified expression 3x -1 when x=4: 3*4 -1=12-1=11. So that seems correct.

But let me check again if my simplification was correct. Because sometimes when you factor or simplify, you might make a mistake.

Original expression:

(3x - 2)(4x + 1) - (3x - 2)4x + 1

Alternatively, maybe I can factor out (3x - 2) from the first two terms?

Let me try that. Let's see:

First two terms: (3x - 2)(4x + 1) - (3x - 2)4x

Factor out (3x - 2):

(3x - 2)[(4x +1) - 4x] +1

Compute inside the brackets: (4x +1 -4x) = 1

So that becomes (3x -2)(1) +1 = 3x -2 +1 = 3x -1. Which is the same as before. So that's another way to see it. So that confirms that the expression simplifies to 3x -1. Therefore, when x=4, it's 11. So that's the answer.

But just to be thorough, let me compute the original expression step by step without simplifying first. Let me do that again with x=4.

Original expression:

(3x - 2)(4x + 1) - (3x - 2)4x + 1

Compute each part:

First part: (3x - 2)(4x + 1). As before, with x=4, that's 10 * 17 = 170.

Second part: -(3x - 2) * 4x. Which is -10 * 16 = -160.

Third part: +1.

So total is 170 -160 +1 = 11. Yep, same result.

Alternatively, if I didn't simplify, but just compute each term:

Alternatively, maybe I can compute the entire expression step by step:

First, compute (3x - 2)(4x + 1):

Let me compute that again with x=4:

(3*4 -2) = 10, (4*4 +1)=17, 10*17=170.

Then subtract (3x -2)*4x: which is 10*16=160. So 170 - 160 = 10.

Then add 1: 10 +1=11. Yep, same answer.

So regardless of the method, the answer is 11. Therefore, I think that's correct.

But just to make sure, let me try plugging x=4 into the simplified expression 3x -1:

3*4 -1 = 12 -1 = 11. Correct.

So, all methods lead to 11. Therefore, the answer is 11.

But just to check if there was any possible misinterpretation of the original expression. For example, if the original expression was meant to be [(3x -2)(4x +1) - (3x -2)4x] +1, which is how I interpreted it, but if someone misreads it as (3x -2)(4x +1 - (3x -2)4x) +1, that would be different. But in that case, the parentheses would be different. Let me check.

Original problem: (3x-2)(4x+1)-(3x-2)4x+1

If someone misreads it as (3x -2)(4x +1 - (3x -2)4x) +1, that would be different. Let me check if that's possible.

But according to order of operations, multiplication is done before addition and subtraction, unless parentheses dictate otherwise. So, the original expression is:

First term: (3x -2)(4x +1)

Second term: - (3x -2) * 4x

Third term: +1

So, the expression is:

First term - second term + third term.

So, unless there are parentheses around the entire second term and third term, which there aren't, so the original expression is as I interpreted.

But just to check, if someone thought that the entire expression is (3x -2)(4x +1 - (3x -2)4x) +1, then that would be different. Let me compute that for x=4, just in case.

Compute inside the parentheses: 4x +1 - (3x -2)4x.

With x=4:

4*4 +1 - (3*4 -2)*4*4 = 16 +1 - (10)*16 = 17 - 160 = -143

Then multiply by (3x -2) which is 10: 10*(-143) = -1430

Then add 1: -1430 +1 = -1429. Which is way different. But that's not the case here, because the original expression doesn't have parentheses around the 4x +1 - (3x -2)4x. So, that's not the correct interpretation.

Therefore, my original interpretation is correct, and the answer is 11.

Alternatively, maybe the original problem had different parentheses? But as written, it's (3x-2)(4x+1)-(3x-2)4x+1. So, no, the way it's written, the expression is as I have.

Therefore, I think the answer is 11. So, I can confidently say that the value is 11 when x=4.

**Final Answer**
The value of the expression when $ x = 4 $ is \boxed{11}.

To evaluate the expression:

$$
(3x - 2)(4x + 1) - (3x - 2) \cdot 4x + 1
$$

when $ x = 4 $, we can either simplify the expression first or substitute $ x = 4 $ directly. Let's proceed step by step.

---

### **Step 1: Simplify the Expression**

We begin by simplifying the expression algebraically.

#### **First Term:**
$$
(3x - 2)(4x + 1)
$$

Use the distributive property (FOIL method):

$$
= 3x \cdot 4x + 3x \cdot 1 - 2 \cdot 4x - 2 \cdot 1
= 12x^2 + 3x - 8x - 2
= 12x^2 - 5x - 2
$$

#### **Second Term:**
$$
-(3x - 2) \cdot 4x
$$

First compute the product:

$$
(3x - 2) \cdot 4x = 12x^2 - 8x
$$

Now apply the negative sign:

$$
- (12x^2 - 8x) = -12x^2 + 8x
$$

#### **Third Term:**
$$
+1
$$

---

### **Step 2: Combine All Terms**

Now combine all the simplified parts:

$$
(12x^2 - 5x - 2) + (-12x^2 + 8x) + 1
$$

Group like terms:

- $ 12x^2 - 12x^2 = 0 $
- $ -5x + 8x = 3x $
- $ -2 + 1 = -1 $

So the entire expression simplifies to:

$$
3x - 1
$$

---

### **Step 3: Substitute $ x = 4 $**

Now plug in $ x = 4 $ into the simplified expression:

$$
3(4) - 1 = 12 - 1 = 11
$$

---

### **Final Answer**

$$
\boxed{11}
$$

Solution:
Since \begin{align*}
(3x-2)(4x+1)-(3x-2)4x+1 &=(3x-2)(4x+1-4x)+1 \\
&=(3x-2) \cdot 1 +1 =3x-1,
\end{align*} when $x=4$ we have the value $3 \cdot 4 -1 =\boxed{11}$.