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, for prime 2: 4 + a must be a multiple of 3.
For prime 3: 1 + b must be a multiple of 3.
For prime 5: 1 + c must be a 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 that aren't already in 240's factorization. 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 4 mod 3? 4 divided by 3 is 1 with remainder 1. So 4 ≡ 1 mod 3. Therefore, 1 + a ≡ 0 mod 3 ⇒ a ≡ -1 mod 3 ⇒ a ≡ 2 mod 3.

So the smallest non-negative integer a that satisfies this is 2. Because if a=2, then 4 + 2 = 6, which is divisible by 3.

Similarly for prime 3: 1 + b ≡ 0 mod 3 ⇒ b ≡ -1 mod 3 ⇒ b ≡ 2 mod 3. So the smallest b is 2.

Same for prime 5: 1 + c ≡ 0 mod 3 ⇒ c ≡ 2 mod 3. So the smallest c is 2.

Therefore, k would be 2² × 3² × 5². Let me compute that.

2² is 4, 3² is 9, 5² is 25. Multiplying them together: 4 × 9 = 36, 36 × 25. Let me calculate that.

36 × 25: Well, 36 × 10 = 360, 36 × 20 = 720, 36 × 25 = 720 + 180 = 900. Wait, no. Wait, 25 is 100/4, so 36 × 25 = 36 × 100 / 4 = 3600 / 4 = 900. Yes.

So k would be 900? But wait, let me check if that's correct.

Wait, if I multiply 240 × 900, is that a perfect cube? Let me check.

First, 240 × 900. Let me compute that.

240 × 900 = 240 × 9 × 100 = 2160 × 100 = 216,000.

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

Well, 60³ is 216,000 because 6³ is 216, and 10³ is 1000, so 6×10=60, so (6×10)³ = 6³ × 10³ = 216 × 1000 = 216,000. Yes! So that works.

But wait, is 900 the smallest possible k? Let me think again.

Is there a smaller k? Because maybe there's a way to have exponents that are multiples of 3 with smaller exponents?

Wait, for each prime, we needed exponents such that when added to the original exponents in 240, they become multiples of 3.

For prime 2: original exponent is 4. The next multiple of 3 greater than or equal to 4 is 6. So 6 - 4 = 2. So we need to add 2.

Similarly for primes 3 and 5: original exponents are 1. The next multiple of 3 is 3, so we need to add 2.

So, that gives exponents of 2 for each of those primes.

But is there a way to have exponents that are higher multiples? For example, if we take 4 + a = 9? Then a would be 5. But that would make k larger. Similarly, for 1 + b = 6, then b=5. So that would make k larger. So the minimal exponents are 2 for each.

Therefore, k is 2² × 3² × 5² = 4 × 9 × 25 = 900.

But wait, let me check if there's a smaller k. Maybe if we have some exponents that are negative? No, because exponents in prime factorization can't be negative. So we have to add exponents.

Alternatively, maybe there's a different combination? For example, if we have some primes with higher exponents? Wait, but we can't have lower exponents because the original exponents are 4, 1, 1.

Wait, for example, for prime 2: 4 + a must be multiple of 3. The possible exponents for a are 2, 5, 8,... So the minimal is 2. Similarly for others.

So, seems like 2² × 3² × 5² is the minimal.

But let me think again. Suppose I have k with exponents such that for each prime, the total exponent is 3. But 240 has exponents 4,1,1. So for prime 2, we need to get to 3? But 4 is already more than 3. Wait, but if we have 4 + a = 3? That would mean a = -1, which is not allowed. So we have to go to the next multiple of 3, which is 6. So 4 + a = 6 ⇒ a = 2.

Similarly for 3 and 5. So that seems correct.

Alternatively, if we have multiple primes? For example, if we have another prime factor in k? But that would require that prime to have exponent multiple of 3, which would make k larger. So no, that's not helpful.

Therefore, k is 2² × 3² × 5² = 900.

But wait, let me check if there's a smaller k. Let me think. Suppose I take k as 2² × 3² × 5² = 900. Is there a smaller number? Let me think.

Suppose I try to find k such that 240k is a cube. Let me think of the cube root of 240k.

But maybe another approach. Let me think of the exponents.

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

To make it a cube, each exponent needs to be a multiple of 3.

So for 2^4: need to multiply by 2^(2) to get 2^6.

For 3^1: multiply by 3^2 to get 3^3.

For 5^1: multiply by 5^2 to get 5^3.

Therefore, k must be 2^2 * 3^2 * 5^2.

So that's 4 * 9 * 25 = 900.

So that seems to be the minimal.

But let me check with smaller numbers. Let me think if there is a smaller k.

Suppose k is 2^2 * 3^2 * 5^2 = 900.

Is there a smaller k? Let me think. For example, if I can find exponents that are lower?

But for each prime, the exponents needed are fixed. For example, for prime 2, you need to have exponent 6. Since original exponent is 4, you need to add 2.

Similarly for 3 and 5.

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

Alternatively, maybe combining exponents? For example, if I have some exponents that are higher? 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 add them, but that would require their exponents to be multiples of 3, which would make k larger.

So, for example, if I take k as 2^2 * 3^2 * 5^2, that's 900. If I tried to use another prime, say 7, then k would be 2^2 * 3^2 * 5^2 * 7^3, which is way bigger.

So, 900 is the minimal.

But let me check with actual numbers. Let me try smaller numbers.

Suppose k is 1. Then 240*1=240. Is 240 a cube? No.

k=2: 240*2=480. Not a cube.

k=3: 720. Not a cube.

k=4: 960. Not a cube.

k=5: 1200. Not a cube.

k=6: 1440. Not a cube.

k=8: 240*8=1920. Not a cube.

k=9: 240*9=2160. Is 2160 a cube? Let me check. 12³ is 1728, 13³ is 2197. So between 12³ and 13³. Not a cube.

k=10: 240*10=2400. Not a cube.

k=12: 240*12=2880. Not a cube.

k=15: 240*15=3600. Not a cube.

k=16: 240*16=3840. Not a cube.

k=18: 240*18=4320. Not a cube.

k=20: 240*20=4800. Not a cube.

k=24: 240*24=5760. Not a cube.

k=25: 240*25=6000. Not a cube.

k=30: 240*30=7200. Not a cube.

k=36: 240*36=8640. Not a cube.

k=40: 240*40=9600. Not a cube.

k=45: 240*45=10800. Not a cube.

k=48: 240*48=11520. Not a cube.

k=50: 240*50=12000. Not a cube.

k=60: 240*60=14400. Not a cube.

k=75: 240*75=18000. Is 18000 a cube? Let me check. 26³ is 17576, 27³ is 19683. So no.

k=80: 240*80=19200. Not a cube.

k=90: 240*90=21600. Not a cube.

k=100: 240*100=24000. Is 24000 a cube? Let me see. 28³ is 21952, 29³ is 24389. So between them. Not a cube.

k=120: 240*120=28800. Not a cube.

k=125: 240*125=30000. Not a cube.

k=150: 240*150=36000. Not a cube.

k=180: 240*180=43200. Not a cube.

k=200: 240*200=48000. Not a cube.

k=225: 240*225=54000. Not a cube.

k=240: 240*240=57600. Not a cube.

k=250: 240*250=60000. Not a cube.

k=300: 240*300=72000. Is 72000 a cube? Let me check. 41³ is 68921, 42³ is 74088. So between them. Not a cube.

k=360: 240*360=86400. Not a cube.

k=400: 240*400=96000. Not a cube.

k=450: 240*450=108000. Is 108000 a cube? Let me check. 47³ is 103823, 48³ is 110592. So between them. Not a cube.

k=500: 240*500=120000. Not a cube.

k=600: 240*600=144000. Is 144000 a cube? Let me check. 52³ is 140608, 53³ is 148877. So between them. Not a cube.

k=720: 240*720=172800. Is that a cube? 55³ is 166375, 56³ is 175616. So between them. Not a cube.

k=900: 240*900=216000. Which is 60³, as I checked earlier.

So, yes, 900 is the first k where 240k is a cube.

Therefore, the minimal k is 900.

But wait, let me check if there is a smaller k that I might have missed. For example, maybe k is not just 2²×3²×5² but something else?

Wait, suppose I think of the cube root of 240k. Let me think.

If 240k is a cube, then 240k = m³ for some integer m.

So, m³ = 240k.

So, 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 = 2⁴ × 3 × 5, then m must be a multiple of 2, 3, and 5.

Let me write m as 2^a × 3^b × 5^c × ...

Then m³ = 2^(3a) × 3^(3b) × 5^(3c) × ...

This must be divisible by 2⁴ × 3¹ × 5¹.

Therefore, 3a ≥ 4, 3b ≥ 1, 3c ≥ 1.

So, solving for a, b, c:

For a: 3a ≥ 4 ⇒ a ≥ 2 (since 3*1=3 <4, 3*2=6 ≥4)

For b: 3b ≥1 ⇒ b ≥1 (since 3*0=0 <1, 3*1=3 ≥1)

For c: same as b, c ≥1.

Therefore, the minimal exponents for a, b, c are 2,1,1.

Therefore, the minimal m is 2² × 3¹ × 5¹ = 4 × 3 × 5 = 60.

Therefore, m=60.

Then, k = m³ / 240 = 60³ / 240.

Compute that: 60³ is 216000. Divided by 240: 216000 / 240.

Divide numerator and denominator by 10: 21600 / 24.

Divide numerator and denominator by 12: 1800 / 2 = 900.

So, k=900.

Therefore, this confirms the earlier result.

So, the minimal k is 900.

Therefore, the answer is 900.

But let me check once more.

Original prime factors of 240: 2⁴ × 3¹ × 5¹.

To make it a cube, we need exponents multiples of 3.

For 2: 4 needs to become 6 (next multiple of 3). So add 2.

For 3: 1 needs to become 3. Add 2.

For 5: same as 3. Add 2.

Therefore, k is 2² × 3² × 5² = 4 × 9 × 25 = 900.

Yes.

Alternatively, if I think of the exponents required for each prime in the product 240k:

For each prime, exponent in 240k is:

- For 2: 4 + a = 3m
- For 3: 1 + b = 3n
- For 5: 1 + c = 3p

Where m, n, p are integers ≥1.

To minimize k, we need to minimize a, b, c.

So, for 2: 4 + a = 3m. The smallest m such that 3m ≥4 is m=2 (since 3*1=3 <4, 3*2=6 ≥4). Therefore, a=6-4=2.

For 3: 1 + b = 3n. The smallest n is 1, so b=3*1 -1=2.

Same for 5: c=2.

Therefore, k=2²×3²×5²=900.

Therefore, the minimal k is 900.

I think that's solid.

But just to make sure, let me check if there's a case where maybe some exponents can be combined or something.

For example, if I have a prime factor that's not in 240, but maybe... but adding new primes would require their exponents to be multiples of 3, which would make k larger.

For example, if I take k=2²×3²×5²×7³, then that would be a multiple of 240, but k is way bigger than 900.

So, no.

Alternatively, if I take k with exponents that are higher multiples? For example, for prime 2, instead of making 4 + a = 6, maybe 4 + a = 9? Then a=5. Then k would be 2^5 × 3^2 × 5^2. Which is 32 × 9 × 25 = 32 × 225 = 7200, which is way bigger than 900.

Same with others. So, no.

Therefore, 900 is indeed the minimal.

Another way to think about it is that the minimal k is the product of primes raised to the power of (3 - (exponent mod 3)) if the exponent mod 3 is not zero, otherwise zero.

Wait, let me explain.

For each prime in the factorization of 240, we check the exponent modulo 3.

For prime 2: exponent is 4. 4 mod 3 is 1. So, to make it 0 mod 3, we need to add (3 - 1) = 2.

For prime 3: exponent is 1. 1 mod 3 is 1. So add 2.

For prime 5: same as 3, add 2.

Therefore, k is 2² × 3² × 5².

Yes, that's another way to think about it.

So, this method also gives 900.

Therefore, I think I can be confident that the minimal k 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 exponents of all primes in its factorization 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 $ 240 \times k $ becomes:

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

To make this a perfect cube, we require:

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

---

### Step 3: Solving the Congruences

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

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

So, the minimal $ k $ is:

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

---

### Step 4: Verifying the Result

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

$$
240 \times 900 = 216000
$$

Now, check if 216000 is a perfect cube:

$$
60^3 = 60 \times 60 \times 60 = 216000
$$

Yes, it is 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, for prime 2: 4 + a must be a multiple of 3.
For prime 3: 1 + b must be a multiple of 3.
For prime 5: 1 + c must be a 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 that aren't already in 240's factorization. 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 4 mod 3? 4 divided by 3 is 1 with remainder 1. So 4 ≡ 1 mod 3. Therefore, 1 + a ≡ 0 mod 3 ⇒ a ≡ -1 mod 3 ⇒ a ≡ 2 mod 3.

So the smallest non-negative integer a that satisfies this is 2. Because if a=2, then 4 + 2 = 6, which is divisible by 3.

Similarly for prime 3: 1 + b ≡ 0 mod 3 ⇒ b ≡ -1 mod 3 ⇒ b ≡ 2 mod 3. So the smallest b is 2.

Same for prime 5: 1 + c ≡ 0 mod 3 ⇒ c ≡ 2 mod 3. So the smallest c is 2.

Therefore, k would be 2² × 3² × 5². Let me compute that.

2² is 4, 3² is 9, 5² is 25. Multiplying them together: 4 × 9 = 36, 36 × 25. Let me calculate that.

36 × 25: Well, 36 × 10 = 360, 36 × 20 = 720, 36 × 25 = 720 + 180 = 900. Wait, no. Wait, 25 is 100/4, so 36 × 25 = 36 × 100 / 4 = 3600 / 4 = 900. Yes.

So k would be 900? But wait, let me check if that's correct.

Wait, if I multiply 240 × 900, is that a perfect cube? Let me check.

First, 240 × 900. Let me compute that.

240 × 900 = 240 × 9 × 100 = 2160 × 100 = 216,000.

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

Well, 60³ is 216,000 because 6³ is 216, and 10³ is 1000, so 6×10=60, so (6×10)³ = 6³ × 10³ = 216 × 1000 = 216,000. Yes! So that works.

But wait, is 900 the smallest possible k? Let me think again.

Is there a smaller k? Because maybe there's a way to have exponents that are multiples of 3 with smaller exponents?

Wait, for each prime, we needed exponents such that when added to the original exponents in 240, they become multiples of 3.

For prime 2: original exponent is 4. The next multiple of 3 greater than or equal to 4 is 6. So 6 - 4 = 2. So we need to add 2.

Similarly for primes 3 and 5: original exponents are 1. The next multiple of 3 is 3, so we need to add 2.

So, that gives exponents of 2 for each of those primes.

But is there a way to have exponents that are higher multiples? For example, if we take 4 + a = 9? Then a would be 5. But that would make k larger. Similarly, for 1 + b = 6, then b=5. So that would make k larger. So the minimal exponents are 2 for each.

Therefore, k is 2² × 3² × 5² = 4 × 9 × 25 = 900.

But wait, let me check if there's a smaller k. Maybe if we have some exponents that are negative? No, because exponents in prime factorization can't be negative. So we have to add exponents.

Alternatively, maybe there's a different combination? For example, if we have some primes with higher exponents? Wait, but we can't have lower exponents because the original exponents are 4, 1, 1.

Wait, for example, for prime 2: 4 + a must be multiple of 3. The possible exponents for a are 2, 5, 8,... So the minimal is 2. Similarly for others.

So, seems like 2² × 3² × 5² is the minimal.

But let me think again. Suppose I have k with exponents such that for each prime, the total exponent is 3. But 240 has exponents 4,1,1. So for prime 2, we need to get to 3? But 4 is already more than 3. Wait, but if we have 4 + a = 3? That would mean a = -1, which is not allowed. So we have to go to the next multiple of 3, which is 6. So 4 + a = 6 ⇒ a = 2.

Similarly for 3 and 5. So that seems correct.

Alternatively, if we have multiple primes? For example, if we have another prime factor in k? But that would require that prime to have exponent multiple of 3, which would make k larger. So no, that's not helpful.

Therefore, k is 2² × 3² × 5² = 900.

But wait, let me check if there's a smaller k. Let me think. Suppose I take k as 2² × 3² × 5² = 900. Is there a smaller number? Let me think.

Suppose I try to find k such that 240k is a cube. Let me think of the cube root of 240k.

But maybe another approach. Let me think of the exponents.

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

To make it a cube, each exponent needs to be a multiple of 3.

So for 2^4: need to multiply by 2^(2) to get 2^6.

For 3^1: multiply by 3^2 to get 3^3.

For 5^1: multiply by 5^2 to get 5^3.

Therefore, k must be 2^2 * 3^2 * 5^2.

So that's 4 * 9 * 25 = 900.

So that seems to be the minimal.

But let me check with smaller numbers. Let me think if there is a smaller k.

Suppose k is 2^2 * 3^2 * 5^2 = 900.

Is there a smaller k? Let me think. For example, if I can find exponents that are lower?

But for each prime, the exponents needed are fixed. For example, for prime 2, you need to have exponent 6. Since original exponent is 4, you need to add 2.

Similarly for 3 and 5.

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

Alternatively, maybe combining exponents? For example, if I have some exponents that are higher? 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 add them, but that would require their exponents to be multiples of 3, which would make k larger.

So, for example, if I take k as 2^2 * 3^2 * 5^2, that's 900. If I tried to use another prime, say 7, then k would be 2^2 * 3^2 * 5^2 * 7^3, which is way bigger.

So, 900 is the minimal.

But let me check with actual numbers. Let me try smaller numbers.

Suppose k is 1. Then 240*1=240. Is 240 a cube? No.

k=2: 240*2=480. Not a cube.

k=3: 720. Not a cube.

k=4: 960. Not a cube.

k=5: 1200. Not a cube.

k=6: 1440. Not a cube.

k=8: 240*8=1920. Not a cube.

k=9: 240*9=2160. Is 2160 a cube? Let me check. 12³ is 1728, 13³ is 2197. So between 12³ and 13³. Not a cube.

k=10: 240*10=2400. Not a cube.

k=12: 240*12=2880. Not a cube.

k=15: 240*15=3600. Not a cube.

k=16: 240*16=3840. Not a cube.

k=18: 240*18=4320. Not a cube.

k=20: 240*20=4800. Not a cube.

k=24: 240*24=5760. Not a cube.

k=25: 240*25=6000. Not a cube.

k=30: 240*30=7200. Not a cube.

k=36: 240*36=8640. Not a cube.

k=40: 240*40=9600. Not a cube.

k=45: 240*45=10800. Not a cube.

k=48: 240*48=11520. Not a cube.

k=50: 240*50=12000. Not a cube.

k=60: 240*60=14400. Not a cube.

k=75: 240*75=18000. Is 18000 a cube? Let me check. 26³ is 17576, 27³ is 19683. So no.

k=80: 240*80=19200. Not a cube.

k=90: 240*90=21600. Not a cube.

k=100: 240*100=24000. Is 24000 a cube? Let me see. 28³ is 21952, 29³ is 24389. So between them. Not a cube.

k=120: 240*120=28800. Not a cube.

k=125: 240*125=30000. Not a cube.

k=150: 240*150=36000. Not a cube.

k=180: 240*180=43200. Not a cube.

k=200: 240*200=48000. Not a cube.

k=225: 240*225=54000. Not a cube.

k=240: 240*240=57600. Not a cube.

k=250: 240*250=60000. Not a cube.

k=300: 240*300=72000. Is 72000 a cube? Let me check. 41³ is 68921, 42³ is 74088. So between them. Not a cube.

k=360: 240*360=86400. Not a cube.

k=400: 240*400=96000. Not a cube.

k=450: 240*450=108000. Is 108000 a cube? Let me check. 47³ is 103823, 48³ is 110592. So between them. Not a cube.

k=500: 240*500=120000. Not a cube.

k=600: 240*600=144000. Is 144000 a cube? Let me check. 52³ is 140608, 53³ is 148877. So between them. Not a cube.

k=720: 240*720=172800. Is that a cube? 55³ is 166375, 56³ is 175616. So between them. Not a cube.

k=900: 240*900=216000. Which is 60³, as I checked earlier.

So, yes, 900 is the first k where 240k is a cube.

Therefore, the minimal k is 900.

But wait, let me check if there is a smaller k that I might have missed. For example, maybe k is not just 2²×3²×5² but something else?

Wait, suppose I think of the cube root of 240k. Let me think.

If 240k is a cube, then 240k = m³ for some integer m.

So, m³ = 240k.

So, 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 = 2⁴ × 3 × 5, then m must be a multiple of 2, 3, and 5.

Let me write m as 2^a × 3^b × 5^c × ...

Then m³ = 2^(3a) × 3^(3b) × 5^(3c) × ...

This must be divisible by 2⁴ × 3¹ × 5¹.

Therefore, 3a ≥ 4, 3b ≥ 1, 3c ≥ 1.

So, solving for a, b, c:

For a: 3a ≥ 4 ⇒ a ≥ 2 (since 3*1=3 <4, 3*2=6 ≥4)

For b: 3b ≥1 ⇒ b ≥1 (since 3*0=0 <1, 3*1=3 ≥1)

For c: same as b, c ≥1.

Therefore, the minimal exponents for a, b, c are 2,1,1.

Therefore, the minimal m is 2² × 3¹ × 5¹ = 4 × 3 × 5 = 60.

Therefore, m=60.

Then, k = m³ / 240 = 60³ / 240.

Compute that: 60³ is 216000. Divided by 240: 216000 / 240.

Divide numerator and denominator by 10: 21600 / 24.

Divide numerator and denominator by 12: 1800 / 2 = 900.

So, k=900.

Therefore, this confirms the earlier result.

So, the minimal k is 900.

Therefore, the answer is 900.

But let me check once more.

Original prime factors of 240: 2⁴ × 3¹ × 5¹.

To make it a cube, we need exponents multiples of 3.

For 2: 4 needs to become 6 (next multiple of 3). So add 2.

For 3: 1 needs to become 3. Add 2.

For 5: same as 3. Add 2.

Therefore, k is 2² × 3² × 5² = 4 × 9 × 25 = 900.

Yes.

Alternatively, if I think of the exponents required for each prime in the product 240k:

For each prime, exponent in 240k is:

- For 2: 4 + a = 3m
- For 3: 1 + b = 3n
- For 5: 1 + c = 3p

Where m, n, p are integers ≥1.

To minimize k, we need to minimize a, b, c.

So, for 2: 4 + a = 3m. The smallest m such that 3m ≥4 is m=2 (since 3*1=3 <4, 3*2=6 ≥4). Therefore, a=6-4=2.

For 3: 1 + b = 3n. The smallest n is 1, so b=3*1 -1=2.

Same for 5: c=2.

Therefore, k=2²×3²×5²=900.

Therefore, the minimal k is 900.

I think that's solid.

But just to make sure, let me check if there's a case where maybe some exponents can be combined or something.

For example, if I have a prime factor that's not in 240, but maybe... but adding new primes would require their exponents to be multiples of 3, which would make k larger.

For example, if I take k=2²×3²×5²×7³, then that would be a multiple of 240, but k is way bigger than 900.

So, no.

Alternatively, if I take k with exponents that are higher multiples? For example, for prime 2, instead of making 4 + a = 6, maybe 4 + a = 9? Then a=5. Then k would be 2^5 × 3^2 × 5^2. Which is 32 × 9 × 25 = 32 × 225 = 7200, which is way bigger than 900.

Same with others. So, no.

Therefore, 900 is indeed the minimal.

Another way to think about it is that the minimal k is the product of primes raised to the power of (3 - (exponent mod 3)) if the exponent mod 3 is not zero, otherwise zero.

Wait, let me explain.

For each prime in the factorization of 240, we check the exponent modulo 3.

For prime 2: exponent is 4. 4 mod 3 is 1. So, to make it 0 mod 3, we need to add (3 - 1) = 2.

For prime 3: exponent is 1. 1 mod 3 is 1. So add 2.

For prime 5: same as 3, add 2.

Therefore, k is 2² × 3² × 5².

Yes, that's another way to think about it.

So, this method also gives 900.

Therefore, I think I can be confident that the minimal k 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 exponents of all primes in its factorization 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 $ 240 \times k $ becomes:

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

To make this a perfect cube, we require:

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

---

### Step 3: Solving the Congruences

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

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

So, the minimal $ k $ is:

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

---

### Step 4: Verifying the Result

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

$$
240 \times 900 = 216000
$$

Now, check if 216000 is a perfect cube:

$$
60^3 = 60 \times 60 \times 60 = 216000
$$

Yes, it is 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. Therefore, the fraction is 2160.5 / 10^7.

Which is 0.00021605.

Yes, same result.

So, 2160.5 divided by 10^7 is moving decimal 7 places to the left.

2160.5 is 2160.5. Moving decimal 7 places left:

Start with 2160.5

1 place: 216.05

2: 21.605

3: 2.1605

4: 0.21605

5: 0.021605

6: 0.0021605

7: 0.00021605

Yes, same result. So, decimal is 0.00021605.

Therefore, the decimal representation is 0.00021605.

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

But wait, the decimal is 0.00021605. So, the digits after the decimal are 0, 0, 0, 2, 1, 6, 0, 5.

But wait, is that all? Let me check again.

Wait, 0.00021605. Let me write it out with all decimal places.

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.

But wait, the question says "the terminating decimal representation". So, does that include leading zeros? For example, is the decimal 0.00021605, which has 8 decimal digits, so the digits are 0, 0, 0, 2, 1, 6, 0, 5.

But when they say "the sum of the digits", do they mean all digits including the leading zeros? Because if you consider the decimal as 0.00021605, then the digits after the decimal are 0, 0, 0, 2, 1, 6, 0, 5. So, sum is 0+0+0+2+1+6+0+5 = 14.

But let me check again. Wait, maybe I made a mistake in the decimal expansion. Let me verify once more.

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

We converted it to 21605 / 10^8, which is 0.00021605.

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

First, compute 5^7. 5^1=5, 5^2=25, 5^3=125, 5^4=625, 5^5=3125, 5^6=15625, 5^7=78125.

Then 2^8 is 256.

So denominator is 78125 * 256. Let me compute that.

First, compute 78125 * 256.

Well, 78125 * 256. Let me note that 256 is 2^8, and 78125 is 5^7.

Alternatively, maybe compute 78125 * 256.

But maybe it's easier to compute 78125 * 256. Let me do that step by step.

First, note that 78125 * 256 = 78125 * (200 + 50 + 6) = 78125*200 + 78125*50 + 78125*6.

But that might take time. Alternatively, note that 78125 * 256 = 78125 * 2^8.

But since we already know that 78125 * 2^8 = 78125 * 256.

Alternatively, since we already converted the fraction to 21605 / 10^8, which is 0.00021605, and we can check if 4321 divided by (5^7 * 2^8) equals 0.00021605.

Alternatively, let me compute 0.00021605 * 5^7 * 2^8 and see if it equals 4321.

First, 0.00021605 * 10^8 = 21605.

Then, 21605 / 5^7 * 2^8? Wait, no. Wait, original fraction is 4321 / (5^7 * 2^8) = 0.00021605.

So, 0.00021605 * (5^7 * 2^8) should be 4321.

But 0.00021605 * 10^8 = 21605.

But 5^7 * 2^8 = 5^7 * 2^7 * 2 = 10^7 * 2.

So, 0.00021605 * 10^7 * 2 = 0.00021605 * 10^7 * 2.

First, 0.00021605 * 10^7 = 2160.5. Then, 2160.5 * 2 = 4321.

Yes! Perfect. So that checks out. Therefore, the decimal is indeed 0.00021605.

Therefore, the decimal representation is 0.00021605.

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

So, the decimal is 0.00021605.

But when writing the decimal, do we include the leading zeros? For example, is the decimal 0.00021605, which has 8 decimal digits, so the digits are 0, 0, 0, 2, 1, 6, 0, 5.

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

So, sum is 0 + 0 + 0 + 2 + 1 + 6 + 0 + 5 = 14.

But let me check again. Wait, maybe there's a mistake here. Let me think.

Wait, the decimal is 0.00021605. So, written out, it's 0.00021605. So, the digits after the decimal are:

Position 1: 0

Position 2: 0

Position 3: 0

Position 4: 2

Position 5: 1

Position 6: 6

Position 7: 0

Position 8: 5

Yes, that's 8 digits. So, adding them up: 0+0+0+2+1+6+0+5 = 14.

But let me check once more. Maybe I missed a digit?

Wait, 0.00021605. Let me write it with all decimal places:

0.00021605

Breaking it down:

- 0.0 (first decimal place)
- 0.00 (second)
- 0.000 (third)
- 0.0002 (fourth)
- 0.00021 (fifth)
- 0.000216 (sixth)
- 0.0002160 (seventh)
- 0.00021605 (eighth)

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

So, sum is 0+0+0+2+1+6+0+5 = 14.

But wait, let me check if there's another way to get the decimal. Maybe I made a mistake in the multiplication earlier?

Wait, original fraction: 4321 / (5^7 * 2^8).

Alternatively, maybe I can compute the decimal by dividing 4321 by (5^7 * 2^8).

But that seems complicated. Alternatively, since we know that 5^7 * 2^8 = 10^7 * 2, so denominator is 2 * 10^7.

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

Which is 2160.5 / 10^7, which is 0.00021605.

Same result.

Alternatively, maybe I can think of 4321 divided by 2^8 * 5^7.

But regardless, the decimal is 0.00021605.

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

Sum is 14.

But wait, let me check once more. Maybe I need to consider more decimal places? Wait, the decimal terminates after 8 digits, so there are no more digits.

Alternatively, maybe I need to write the decimal as 0.00021605, which is 0.00021605.

But let me check if there's a possibility that the decimal has more digits? For example, if there was a remainder?

But since we converted the fraction to have denominator 10^8, which is a power of 10, the decimal must terminate exactly at 8 digits.

Therefore, there are no more digits.

Therefore, the digits are as above.

But let me check 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. Let me note that 78125 * 256 = 78125 * (2^8).

Alternatively, since 78125 * 2 = 156250

78125 * 4 = 312500

78125 * 8 = 625000

78125 * 16 = 1,250,000

78125 * 32 = 2,500,000

78125 * 64 = 5,000,000

78125 * 128 = 10,000,000

78125 * 256 = 20,000,000

Wait, that seems too easy. Wait, 78125 * 2^8 = 78125 * 256.

But 78125 * 2^8 = 78125 * 256.

But 78125 * 2^8 = 78125 * 2^3 * 2^5 = 78125 * 8 * 32.

But 78125 * 8 = 625,000. Then 625,000 * 32.

Compute 625,000 * 32:

Well, 625,000 * 30 = 18,750,000

625,000 * 2 = 1,250,000

Total: 18,750,000 + 1,250,000 = 20,000,000.

So, denominator is 20,000,000.

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

So, 4321 divided by 20,000,000.

Which is equal to (4321 / 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.

Same result.

Therefore, decimal is 0.00021605.

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

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

But wait, let me check once more. Maybe I need to consider that the decimal is 0.00021605, which is 0.00021605.

But if I write this as a decimal, is there a possibility that there are more digits? For example, if there was a remainder?

But since we have denominator as 10^8, the decimal terminates exactly at 8 digits.

Alternatively, maybe I need to check if 4321 * 5 is indeed 21605.

Yes, 4321 * 5:

4000*5=20,000

300*5=1,500

20*5=100

1*5=5

Total: 20,000 + 1,500 = 21,500; 21,500 + 100 = 21,600; 21,600 + 5 = 21,605.

Yes, correct.

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

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

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

But wait, let me check once more. Maybe I made a mistake in the decimal places?

Wait, 21,605 divided by 10^8.

Let me think of 21,605 as 21605.

Divided by 10^8 is moving decimal 8 places left.

So, starting from 21605.

Original number: 21605.0

After moving decimal 1 place: 2160.50

After 2: 216.050

After 3: 21.6050

After 4: 2.16050

After 5: 0.216050

After 6: 0.0216050

After 7: 0.00216050

After 8: 0.000216050

So, the decimal is 0.000216050. But since it's terminating, the last zero can be omitted. However, when considering the decimal representation, it's 0.00021605, with the last digit being 5.

But in terms of digits, even if there's a trailing zero, it's still part of the decimal. However, in this case, after moving the decimal 8 places, we have 0.00021605, which is 8 decimal digits.

But let me check if there's a trailing zero.

Wait, 21,605 divided by 10^8 is 0.00021605.

But 21,605 is 21605, which is 5 digits. So, when you divide by 10^8, you have 8 decimal places. So, the decimal is 0.00021605.

But if you write it as 0.00021605, then the digits are 0, 0, 0, 2, 1, 6, 0, 5.

But if you write it as 0.000216050, then there's an extra zero, but since it's terminating, the trailing zero is not necessary. However, the question says "the terminating decimal representation".

In decimal representations, trailing zeros after the last non-zero digit are usually omitted unless specified. However, when considering the exact decimal representation, it's important to note that the decimal terminates at the 8th digit.

But in reality, 21,605 / 10^8 is exactly equal to 0.00021605, with no trailing zeros. Because 21,605 is 21605, which is 5 digits, so when you divide by 10^8, you have 8 decimal places, which would be 0.00021605.

Wait, but 21,605 is 21605. So, if you have 21605 divided by 10^8, you can think of it as 0.00021605.

But let me check with another example. For example, 1 divided by 10^3 is 0.001. So, three decimal places. Similarly, 123 divided by 10^5 is 0.00123. So, five decimal places.

Therefore, 21605 divided by 10^8 is 0.00021605.

So, the decimal is 0.00021605.

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

Therefore, sum is 0+0+0+2+1+6+0+5 = 14.

But let me check once more. Maybe I need to consider that the decimal is 0.00021605, which is 0.00021605.

But if I write this out, the digits are:

- 0 (1st)
- 0 (2nd)
- 0 (3rd)
- 2 (4th)
- 1 (5th)
- 6 (6th)
- 0 (7th)
- 5 (8th)

Yes, that's 8 digits.

So, adding them up: 0 + 0 + 0 + 2 + 1 + 6 + 0 + 5 = 14.

But wait, I feel like 14 is a small number. Let me check if there's a mistake in the decimal expansion.

Wait, let me think of another way. Let me compute 4321 divided by 5^7 * 2^8.

But 5^7 * 2^8 = 5^7 * 2^7 * 2 = 10^7 * 2.

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

Yes, same result.

Alternatively, maybe I can compute 4321 divided by 2^8 first, then divided by 5^7.

But 2^8 is 256.

So, 4321 divided by 256. Let me compute that.

Compute 4321 ÷ 256.

Well, 256 * 16 = 4096.

Subtract 4096 from 4321: 4321 - 4096 = 225.

So, 4321 / 256 = 16 + 225/256.

Then, divide that by 5^7.

So, 16 + 225/256 divided by 78125.

But this seems more complicated.

Alternatively, maybe convert 225/256 to decimal.

225 divided by 256.

Well, 225/256 is approximately 0.87890625.

Therefore, 16.87890625 divided by 78125.

But this seems messy.

Alternatively, maybe not helpful.

But since we already have the decimal as 0.00021605, and verified it multiple times, I think that is correct.

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

Sum is 14.

But let me check once more. Maybe I made a mistake in the multiplication earlier?

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

Multiply numerator and denominator by 5^1 to get denominator as 10^8.

Numerator becomes 4321 * 5 = 21605.

Therefore, decimal is 21605 / 10^8 = 0.00021605.

Yes.

Alternatively, maybe I can write 21605 as 21605.00000000 and divide by 10^8.

But that would be moving decimal 8 places left, resulting in 0.00021605.

Yes.

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

Sum is 14.

But wait, let me check with another example. Suppose I have 1/10^8 = 0.00000001. Sum of digits is 1.

Another example: 123/10^5 = 0.00123. Sum of digits is 0+0+1+2+3=6.

Similarly, 21605/10^8 = 0.00021605. Sum of digits is 0+0+0+2+1+6+0+5=14.

Therefore, seems correct.

But maybe the question is expecting something else?

Wait, the problem says "the sum of the digits in the terminating decimal representation".

But sometimes, when people refer to the decimal representation, they might consider the entire number, including the integer part. But in this case, the decimal is 0.00021605, so the integer part is 0, and the decimal part is 0.00021605.

But the question says "the sum of the digits in the terminating decimal representation". So, does that include the 0 before the decimal?

Wait, 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 number, including the integer part, then the digits are 0, 0, 0, 2, 1, 6, 0, 5.

But if we consider only the decimal part, then it's 0,0,0,2,1,6,0,5.

Either way, the sum is 14.

But maybe the question is considering the decimal part only?

But the problem says "the terminating decimal representation", which is 0.00021605. So, the digits are 0, 0, 0, 2, 1, 6, 0, 5.

But if you consider the entire number, including the leading zero before the decimal, then the digits are 0, 0, 0, 2, 1, 6, 0, 5.

But in reality, the number is 0.00021605, so the digits are as above.

Therefore, the sum is 14.

But let me check once more. Maybe I made a mistake in the decimal places?

Wait, 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

Yes, 8 digits.

So, adding them up: 0+0+0+2+1+6+0+5 = 14.

Therefore, the answer is 14.

But just to be thorough, let me check if there's an alternative interpretation.

Suppose someone thought that the decimal is 0.00021605, but maybe they consider the digits as 0, 0, 0, 2, 1, 6, 0, 5, which is 8 digits.

Alternatively, if there was a miscalculation in the decimal expansion.

Wait, let me check with another method. Let me compute 4321 divided by 5^7 * 2^8.

First, compute 5^7 = 78125, 2^8 = 256.

Multiply them: 78125 * 256.

As before, we found that 78125 * 256 = 20,000,000.

Therefore, 4321 / 20,000,000.

Let me compute this division.

Divide 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 = 3,210.

Wait, this seems messy. Maybe another approach.

Alternatively, since we know that 4321 / 20,000,000 = 0.00021605, as established before.

Alternatively, think of 4321 / 20,000,000 = (4321 / 20) / 1,000,000 = 216.05 / 1,000,000 = 0.00021605.

Yes.

Therefore, decimal is 0.00021605.

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

Sum is 14.

Therefore, I think the answer is 14.

But just to be 100% sure, let me check with a calculator.

But since I don't have a calculator here, I can do the following.

Let me compute 0.00021605 * 20,000,000.

0.00021605 * 20,000,000.

First, 0.00021605 * 10,000,000 = 2,160.5

Then, 0.00021605 * 10,000,000 = another 2,160.5

Total: 2,160.5 + 2,160.5 = 4,321.

Yes! Perfect. Therefore, 0.00021605 * 20,000,000 = 4,321.

Therefore, the decimal is correct.

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

Sum is 14.

Therefore, 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:

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

We can rewrite this as:

$$
(5^7 \cdot 2^7) \cdot 2 = 10^7 \cdot 2
$$

So, the denominator is $ 10^7 \cdot 2 $, which is not a power of 10. However, to make the denominator a power of 10, we can multiply both numerator and denominator by $ 5 $, so that:

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

Now the denominator is $ 10^8 $, and the fraction becomes:

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

---

### Step 2: Convert to Decimal Form

Dividing by $ 10^8 $ is equivalent to moving the decimal point 8 places to the left in the numerator:

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

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

---

### Step 3: Identify the Digits and Compute the Sum

The decimal is:

$$
0.00021605
$$

The digits after the decimal point are:

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

Now, sum these 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. Therefore, the fraction is 2160.5 / 10^7.

Which is 0.00021605.

Yes, same result.

So, 2160.5 divided by 10^7 is moving decimal 7 places to the left.

2160.5 is 2160.5. Moving decimal 7 places left:

Start with 2160.5

1 place: 216.05

2: 21.605

3: 2.1605

4: 0.21605

5: 0.021605

6: 0.0021605

7: 0.00021605

Yes, same result. So, decimal is 0.00021605.

Therefore, the decimal representation is 0.00021605.

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

But wait, the decimal is 0.00021605. So, the digits after the decimal are 0, 0, 0, 2, 1, 6, 0, 5.

But wait, is that all? Let me check again.

Wait, 0.00021605. Let me write it out with all decimal places.

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.

But wait, the question says "the terminating decimal representation". So, does that include leading zeros? For example, is the decimal 0.00021605, which has 8 decimal digits, so the digits are 0, 0, 0, 2, 1, 6, 0, 5.

But when they say "the sum of the digits", do they mean all digits including the leading zeros? Because if you consider the decimal as 0.00021605, then the digits after the decimal are 0, 0, 0, 2, 1, 6, 0, 5. So, sum is 0+0+0+2+1+6+0+5 = 14.

But let me check again. Wait, maybe I made a mistake in the decimal expansion. Let me verify once more.

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

We converted it to 21605 / 10^8, which is 0.00021605.

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

First, compute 5^7. 5^1=5, 5^2=25, 5^3=125, 5^4=625, 5^5=3125, 5^6=15625, 5^7=78125.

Then 2^8 is 256.

So denominator is 78125 * 256. Let me compute that.

First, compute 78125 * 256.

Well, 78125 * 256. Let me note that 256 is 2^8, and 78125 is 5^7.

Alternatively, maybe compute 78125 * 256.

But maybe it's easier to compute 78125 * 256. Let me do that step by step.

First, note that 78125 * 256 = 78125 * (200 + 50 + 6) = 78125*200 + 78125*50 + 78125*6.

But that might take time. Alternatively, note that 78125 * 256 = 78125 * 2^8.

But since we already know that 78125 * 2^8 = 78125 * 256.

Alternatively, since we already converted the fraction to 21605 / 10^8, which is 0.00021605, and we can check if 4321 divided by (5^7 * 2^8) equals 0.00021605.

Alternatively, let me compute 0.00021605 * 5^7 * 2^8 and see if it equals 4321.

First, 0.00021605 * 10^8 = 21605.

Then, 21605 / 5^7 * 2^8? Wait, no. Wait, original fraction is 4321 / (5^7 * 2^8) = 0.00021605.

So, 0.00021605 * (5^7 * 2^8) should be 4321.

But 0.00021605 * 10^8 = 21605.

But 5^7 * 2^8 = 5^7 * 2^7 * 2 = 10^7 * 2.

So, 0.00021605 * 10^7 * 2 = 0.00021605 * 10^7 * 2.

First, 0.00021605 * 10^7 = 2160.5. Then, 2160.5 * 2 = 4321.

Yes! Perfect. So that checks out. Therefore, the decimal is indeed 0.00021605.

Therefore, the decimal representation is 0.00021605.

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

So, the decimal is 0.00021605.

But when writing the decimal, do we include the leading zeros? For example, is the decimal 0.00021605, which has 8 decimal digits, so the digits are 0, 0, 0, 2, 1, 6, 0, 5.

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

So, sum is 0 + 0 + 0 + 2 + 1 + 6 + 0 + 5 = 14.

But let me check again. Wait, maybe there's a mistake here. Let me think.

Wait, the decimal is 0.00021605. So, written out, it's 0.00021605. So, the digits after the decimal are:

Position 1: 0

Position 2: 0

Position 3: 0

Position 4: 2

Position 5: 1

Position 6: 6

Position 7: 0

Position 8: 5

Yes, that's 8 digits. So, adding them up: 0+0+0+2+1+6+0+5 = 14.

But let me check once more. Maybe I missed a digit?

Wait, 0.00021605. Let me write it with all decimal places:

0.00021605

Breaking it down:

- 0.0 (first decimal place)
- 0.00 (second)
- 0.000 (third)
- 0.0002 (fourth)
- 0.00021 (fifth)
- 0.000216 (sixth)
- 0.0002160 (seventh)
- 0.00021605 (eighth)

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

So, sum is 0+0+0+2+1+6+0+5 = 14.

But wait, let me check if there's another way to get the decimal. Maybe I made a mistake in the multiplication earlier?

Wait, original fraction: 4321 / (5^7 * 2^8).

Alternatively, maybe I can compute the decimal by dividing 4321 by (5^7 * 2^8).

But that seems complicated. Alternatively, since we know that 5^7 * 2^8 = 10^7 * 2, so denominator is 2 * 10^7.

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

Which is 2160.5 / 10^7, which is 0.00021605.

Same result.

Alternatively, maybe I can think of 4321 divided by 2^8 * 5^7.

But regardless, the decimal is 0.00021605.

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

Sum is 14.

But wait, let me check once more. Maybe I need to consider more decimal places? Wait, the decimal terminates after 8 digits, so there are no more digits.

Alternatively, maybe I need to write the decimal as 0.00021605, which is 0.00021605.

But let me check if there's a possibility that the decimal has more digits? For example, if there was a remainder?

But since we converted the fraction to have denominator 10^8, which is a power of 10, the decimal must terminate exactly at 8 digits.

Therefore, there are no more digits.

Therefore, the digits are as above.

But let me check 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. Let me note that 78125 * 256 = 78125 * (2^8).

Alternatively, since 78125 * 2 = 156250

78125 * 4 = 312500

78125 * 8 = 625000

78125 * 16 = 1,250,000

78125 * 32 = 2,500,000

78125 * 64 = 5,000,000

78125 * 128 = 10,000,000

78125 * 256 = 20,000,000

Wait, that seems too easy. Wait, 78125 * 2^8 = 78125 * 256.

But 78125 * 2^8 = 78125 * 256.

But 78125 * 2^8 = 78125 * 2^3 * 2^5 = 78125 * 8 * 32.

But 78125 * 8 = 625,000. Then 625,000 * 32.

Compute 625,000 * 32:

Well, 625,000 * 30 = 18,750,000

625,000 * 2 = 1,250,000

Total: 18,750,000 + 1,250,000 = 20,000,000.

So, denominator is 20,000,000.

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

So, 4321 divided by 20,000,000.

Which is equal to (4321 / 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.

Same result.

Therefore, decimal is 0.00021605.

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

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

But wait, let me check once more. Maybe I need to consider that the decimal is 0.00021605, which is 0.00021605.

But if I write this as a decimal, is there a possibility that there are more digits? For example, if there was a remainder?

But since we have denominator as 10^8, the decimal terminates exactly at 8 digits.

Alternatively, maybe I need to check if 4321 * 5 is indeed 21605.

Yes, 4321 * 5:

4000*5=20,000

300*5=1,500

20*5=100

1*5=5

Total: 20,000 + 1,500 = 21,500; 21,500 + 100 = 21,600; 21,600 + 5 = 21,605.

Yes, correct.

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

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

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

But wait, let me check once more. Maybe I made a mistake in the decimal places?

Wait, 21,605 divided by 10^8.

Let me think of 21,605 as 21605.

Divided by 10^8 is moving decimal 8 places left.

So, starting from 21605.

Original number: 21605.0

After moving decimal 1 place: 2160.50

After 2: 216.050

After 3: 21.6050

After 4: 2.16050

After 5: 0.216050

After 6: 0.0216050

After 7: 0.00216050

After 8: 0.000216050

So, the decimal is 0.000216050. But since it's terminating, the last zero can be omitted. However, when considering the decimal representation, it's 0.00021605, with the last digit being 5.

But in terms of digits, even if there's a trailing zero, it's still part of the decimal. However, in this case, after moving the decimal 8 places, we have 0.00021605, which is 8 decimal digits.

But let me check if there's a trailing zero.

Wait, 21,605 divided by 10^8 is 0.00021605.

But 21,605 is 21605, which is 5 digits. So, when you divide by 10^8, you have 8 decimal places. So, the decimal is 0.00021605.

But if you write it as 0.00021605, then the digits are 0, 0, 0, 2, 1, 6, 0, 5.

But if you write it as 0.000216050, then there's an extra zero, but since it's terminating, the trailing zero is not necessary. However, the question says "the terminating decimal representation".

In decimal representations, trailing zeros after the last non-zero digit are usually omitted unless specified. However, when considering the exact decimal representation, it's important to note that the decimal terminates at the 8th digit.

But in reality, 21,605 / 10^8 is exactly equal to 0.00021605, with no trailing zeros. Because 21,605 is 21605, which is 5 digits, so when you divide by 10^8, you have 8 decimal places, which would be 0.00021605.

Wait, but 21,605 is 21605. So, if you have 21605 divided by 10^8, you can think of it as 0.00021605.

But let me check with another example. For example, 1 divided by 10^3 is 0.001. So, three decimal places. Similarly, 123 divided by 10^5 is 0.00123. So, five decimal places.

Therefore, 21605 divided by 10^8 is 0.00021605.

So, the decimal is 0.00021605.

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

Therefore, sum is 0+0+0+2+1+6+0+5 = 14.

But let me check once more. Maybe I need to consider that the decimal is 0.00021605, which is 0.00021605.

But if I write this out, the digits are:

- 0 (1st)
- 0 (2nd)
- 0 (3rd)
- 2 (4th)
- 1 (5th)
- 6 (6th)
- 0 (7th)
- 5 (8th)

Yes, that's 8 digits.

So, adding them up: 0 + 0 + 0 + 2 + 1 + 6 + 0 + 5 = 14.

But wait, I feel like 14 is a small number. Let me check if there's a mistake in the decimal expansion.

Wait, let me think of another way. Let me compute 4321 divided by 5^7 * 2^8.

But 5^7 * 2^8 = 5^7 * 2^7 * 2 = 10^7 * 2.

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

Yes, same result.

Alternatively, maybe I can compute 4321 divided by 2^8 first, then divided by 5^7.

But 2^8 is 256.

So, 4321 divided by 256. Let me compute that.

Compute 4321 ÷ 256.

Well, 256 * 16 = 4096.

Subtract 4096 from 4321: 4321 - 4096 = 225.

So, 4321 / 256 = 16 + 225/256.

Then, divide that by 5^7.

So, 16 + 225/256 divided by 78125.

But this seems more complicated.

Alternatively, maybe convert 225/256 to decimal.

225 divided by 256.

Well, 225/256 is approximately 0.87890625.

Therefore, 16.87890625 divided by 78125.

But this seems messy.

Alternatively, maybe not helpful.

But since we already have the decimal as 0.00021605, and verified it multiple times, I think that is correct.

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

Sum is 14.

But let me check once more. Maybe I made a mistake in the multiplication earlier?

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

Multiply numerator and denominator by 5^1 to get denominator as 10^8.

Numerator becomes 4321 * 5 = 21605.

Therefore, decimal is 21605 / 10^8 = 0.00021605.

Yes.

Alternatively, maybe I can write 21605 as 21605.00000000 and divide by 10^8.

But that would be moving decimal 8 places left, resulting in 0.00021605.

Yes.

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

Sum is 14.

But wait, let me check with another example. Suppose I have 1/10^8 = 0.00000001. Sum of digits is 1.

Another example: 123/10^5 = 0.00123. Sum of digits is 0+0+1+2+3=6.

Similarly, 21605/10^8 = 0.00021605. Sum of digits is 0+0+0+2+1+6+0+5=14.

Therefore, seems correct.

But maybe the question is expecting something else?

Wait, the problem says "the sum of the digits in the terminating decimal representation".

But sometimes, when people refer to the decimal representation, they might consider the entire number, including the integer part. But in this case, the decimal is 0.00021605, so the integer part is 0, and the decimal part is 0.00021605.

But the question says "the sum of the digits in the terminating decimal representation". So, does that include the 0 before the decimal?

Wait, 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 number, including the integer part, then the digits are 0, 0, 0, 2, 1, 6, 0, 5.

But if we consider only the decimal part, then it's 0,0,0,2,1,6,0,5.

Either way, the sum is 14.

But maybe the question is considering the decimal part only?

But the problem says "the terminating decimal representation", which is 0.00021605. So, the digits are 0, 0, 0, 2, 1, 6, 0, 5.

But if you consider the entire number, including the leading zero before the decimal, then the digits are 0, 0, 0, 2, 1, 6, 0, 5.

But in reality, the number is 0.00021605, so the digits are as above.

Therefore, the sum is 14.

But let me check once more. Maybe I made a mistake in the decimal places?

Wait, 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

Yes, 8 digits.

So, adding them up: 0+0+0+2+1+6+0+5 = 14.

Therefore, the answer is 14.

But just to be thorough, let me check if there's an alternative interpretation.

Suppose someone thought that the decimal is 0.00021605, but maybe they consider the digits as 0, 0, 0, 2, 1, 6, 0, 5, which is 8 digits.

Alternatively, if there was a miscalculation in the decimal expansion.

Wait, let me check with another method. Let me compute 4321 divided by 5^7 * 2^8.

First, compute 5^7 = 78125, 2^8 = 256.

Multiply them: 78125 * 256.

As before, we found that 78125 * 256 = 20,000,000.

Therefore, 4321 / 20,000,000.

Let me compute this division.

Divide 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 = 3,210.

Wait, this seems messy. Maybe another approach.

Alternatively, since we know that 4321 / 20,000,000 = 0.00021605, as established before.

Alternatively, think of 4321 / 20,000,000 = (4321 / 20) / 1,000,000 = 216.05 / 1,000,000 = 0.00021605.

Yes.

Therefore, decimal is 0.00021605.

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

Sum is 14.

Therefore, I think the answer is 14.

But just to be 100% sure, let me check with a calculator.

But since I don't have a calculator here, I can do the following.

Let me compute 0.00021605 * 20,000,000.

0.00021605 * 20,000,000.

First, 0.00021605 * 10,000,000 = 2,160.5

Then, 0.00021605 * 10,000,000 = another 2,160.5

Total: 2,160.5 + 2,160.5 = 4,321.

Yes! Perfect. Therefore, 0.00021605 * 20,000,000 = 4,321.

Therefore, the decimal is correct.

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

Sum is 14.

Therefore, 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:

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

We can rewrite this as:

$$
(5^7 \cdot 2^7) \cdot 2 = 10^7 \cdot 2
$$

So, the denominator is $ 10^7 \cdot 2 $, which is not a power of 10. However, to make the denominator a power of 10, we can multiply both numerator and denominator by $ 5 $, so that:

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

Now the denominator is $ 10^8 $, and the fraction becomes:

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

---

### Step 2: Convert to Decimal Form

Dividing by $ 10^8 $ is equivalent to moving the decimal point 8 places to the left in the numerator:

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

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

---

### Step 3: Identify the Digits and Compute the Sum

The decimal is:

$$
0.00021605
$$

The digits after the decimal point are:

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

Now, sum these 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

Is that right? Let me confirm. The original problem is written as:

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

So, the first term is (3x - 2)(4x + 1), then subtract (3x - 2) times 4x, and then add 1. So yes, that's how it's structured. So, maybe I can simplify this expression before plugging in x = 4? Let me try that.

First, let me look at the first two terms: (3x - 2)(4x + 1) - (3x - 2)(4x). Maybe I can factor out (3x - 2) from both terms? Let me try that.

So, factoring (3x - 2) from the first two terms:

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

Let me compute the part inside the brackets:

(4x + 1) - 4x = 4x + 1 - 4x = 1

Oh! That simplifies nicely. So then the entire expression becomes:

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

Wait, so the entire original expression simplifies to 3x - 1? That seems too simple. Let me check my steps again.

Original expression:

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

Factor out (3x - 2) from the first two terms:

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

Yes, that's correct. Then inside the brackets:

(4x + 1) - 4x = 1. So, that gives:

(3x - 2)(1) + 1 = 3x - 2 + 1 = 3x - 1. Okay, that seems correct. So, if that's the case, then substituting x = 4 would be straightforward.

So, 3x - 1 when x = 4 is 3*4 - 1 = 12 - 1 = 11. Therefore, the value is 11. But let me verify this by plugging x = 4 into the original expression directly, just to make sure I didn't make a mistake in simplifying.

Original expression:

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

Let me compute each part step by step when x = 4.

First, compute 3x - 2 when x = 4:

3*4 - 2 = 12 - 2 = 10

Then compute 4x + 1 when x = 4:

4*4 + 1 = 16 + 1 = 17

So, the first term is (10)(17) = 170

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

First, 3x - 2 is 10, as above. Then 4x is 16. So, -(10)(16) = -160

Third term is +1.

So, putting it all together: 170 - 160 + 1 = (170 - 160) + 1 = 10 + 1 = 11. Okay, that matches the simplified expression. So, the answer is 11. Therefore, my simplification was correct.

Alternatively, maybe I can check by expanding the original expression without factoring. Let me try that as another method to confirm.

Original expression:

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

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

Multiply 3x by 4x: 12x²

Multiply 3x by 1: 3x

Multiply -2 by 4x: -8x

Multiply -2 by 1: -2

So, altogether: 12x² + 3x - 8x - 2 = 12x² - 5x - 2

Then, expand -(3x - 2)(4x):

First, compute (3x - 2)(4x) = 12x² - 8x, then apply the negative sign: -12x² + 8x

Then, add the +1 at the end.

So, putting all together:

First part: 12x² - 5x - 2

Second part: -12x² + 8x

Third part: +1

Combine them:

12x² - 5x - 2 -12x² + 8x + 1

Combine like terms:

12x² -12x² = 0

-5x + 8x = 3x

-2 + 1 = -1

So overall, 3x - 1. Which is the same as before. Therefore, the simplified expression is indeed 3x - 1, so substituting x = 4 gives 11. So, that's consistent.

Therefore, regardless of the method, the answer is 11. Therefore, I can be confident that the value is 11 when x = 4.

But just to make sure, let me do the entire calculation again step by step with x = 4 in the original expression.

Original expression:

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

Compute each part:

First term: (3*4 - 2)(4*4 + 1) = (12 - 2)(16 + 1) = (10)(17) = 170

Second term: -(3*4 - 2)*4*4 = -(10)*16 = -160

Third term: +1

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

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

First term: 170

Second term: - (3x - 2)*4x. Let me compute (3x - 2)*4x first. 3x - 2 is 10, 4x is 16, so 10*16 = 160, then with the negative sign, it's -160.

Third term is +1. So 170 - 160 + 1 = 11. Yep.

Alternatively, maybe I can think of the original expression as:

Let me denote A = (3x - 2). Then the expression is A*(4x + 1) - A*(4x) + 1. Which is A*(4x + 1 - 4x) + 1 = A*(1) + 1 = A + 1. Since A = 3x - 2, then A + 1 = 3x - 2 + 1 = 3x - 1. So that's another way to see it. So, regardless of the approach, it's 3x - 1. Therefore, substituting x = 4 gives 11.

Therefore, I think the answer is definitely 11. I don't see any mistakes in my reasoning. All methods lead to the same result. So, I can be confident 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:

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

when $ x = 4 $, we can simplify the expression first before substituting the value of $ x $.

---

### Step 1: Factor the expression

We notice that both the first and second terms share a common factor of $ (3x - 2) $. So we can factor it out:

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

Now simplify the expression inside the brackets:

$$
(4x + 1) - 4x = 1
$$

So the expression becomes:

$$
(3x - 2)(1) + 1 = (3x - 2) + 1 = 3x - 1
$$

---

### Step 2: Substitute $ x = 4 $

Now that the expression is simplified to $ 3x - 1 $, we substitute $ x = 4 $:

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

---

### 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

Is that right? Let me confirm. The original problem is written as:

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

So, the first term is (3x - 2)(4x + 1), then subtract (3x - 2) times 4x, and then add 1. So yes, that's how it's structured. So, maybe I can simplify this expression before plugging in x = 4? Let me try that.

First, let me look at the first two terms: (3x - 2)(4x + 1) - (3x - 2)(4x). Maybe I can factor out (3x - 2) from both terms? Let me try that.

So, factoring (3x - 2) from the first two terms:

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

Let me compute the part inside the brackets:

(4x + 1) - 4x = 4x + 1 - 4x = 1

Oh! That simplifies nicely. So then the entire expression becomes:

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

Wait, so the entire original expression simplifies to 3x - 1? That seems too simple. Let me check my steps again.

Original expression:

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

Factor out (3x - 2) from the first two terms:

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

Yes, that's correct. Then inside the brackets:

(4x + 1) - 4x = 1. So, that gives:

(3x - 2)(1) + 1 = 3x - 2 + 1 = 3x - 1. Okay, that seems correct. So, if that's the case, then substituting x = 4 would be straightforward.

So, 3x - 1 when x = 4 is 3*4 - 1 = 12 - 1 = 11. Therefore, the value is 11. But let me verify this by plugging x = 4 into the original expression directly, just to make sure I didn't make a mistake in simplifying.

Original expression:

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

Let me compute each part step by step when x = 4.

First, compute 3x - 2 when x = 4:

3*4 - 2 = 12 - 2 = 10

Then compute 4x + 1 when x = 4:

4*4 + 1 = 16 + 1 = 17

So, the first term is (10)(17) = 170

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

First, 3x - 2 is 10, as above. Then 4x is 16. So, -(10)(16) = -160

Third term is +1.

So, putting it all together: 170 - 160 + 1 = (170 - 160) + 1 = 10 + 1 = 11. Okay, that matches the simplified expression. So, the answer is 11. Therefore, my simplification was correct.

Alternatively, maybe I can check by expanding the original expression without factoring. Let me try that as another method to confirm.

Original expression:

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

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

Multiply 3x by 4x: 12x²

Multiply 3x by 1: 3x

Multiply -2 by 4x: -8x

Multiply -2 by 1: -2

So, altogether: 12x² + 3x - 8x - 2 = 12x² - 5x - 2

Then, expand -(3x - 2)(4x):

First, compute (3x - 2)(4x) = 12x² - 8x, then apply the negative sign: -12x² + 8x

Then, add the +1 at the end.

So, putting all together:

First part: 12x² - 5x - 2

Second part: -12x² + 8x

Third part: +1

Combine them:

12x² - 5x - 2 -12x² + 8x + 1

Combine like terms:

12x² -12x² = 0

-5x + 8x = 3x

-2 + 1 = -1

So overall, 3x - 1. Which is the same as before. Therefore, the simplified expression is indeed 3x - 1, so substituting x = 4 gives 11. So, that's consistent.

Therefore, regardless of the method, the answer is 11. Therefore, I can be confident that the value is 11 when x = 4.

But just to make sure, let me do the entire calculation again step by step with x = 4 in the original expression.

Original expression:

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

Compute each part:

First term: (3*4 - 2)(4*4 + 1) = (12 - 2)(16 + 1) = (10)(17) = 170

Second term: -(3*4 - 2)*4*4 = -(10)*16 = -160

Third term: +1

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

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

First term: 170

Second term: - (3x - 2)*4x. Let me compute (3x - 2)*4x first. 3x - 2 is 10, 4x is 16, so 10*16 = 160, then with the negative sign, it's -160.

Third term is +1. So 170 - 160 + 1 = 11. Yep.

Alternatively, maybe I can think of the original expression as:

Let me denote A = (3x - 2). Then the expression is A*(4x + 1) - A*(4x) + 1. Which is A*(4x + 1 - 4x) + 1 = A*(1) + 1 = A + 1. Since A = 3x - 2, then A + 1 = 3x - 2 + 1 = 3x - 1. So that's another way to see it. So, regardless of the approach, it's 3x - 1. Therefore, substituting x = 4 gives 11.

Therefore, I think the answer is definitely 11. I don't see any mistakes in my reasoning. All methods lead to the same result. So, I can be confident 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:

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

when $ x = 4 $, we can simplify the expression first before substituting the value of $ x $.

---

### Step 1: Factor the expression

We notice that both the first and second terms share a common factor of $ (3x - 2) $. So we can factor it out:

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

Now simplify the expression inside the brackets:

$$
(4x + 1) - 4x = 1
$$

So the expression becomes:

$$
(3x - 2)(1) + 1 = (3x - 2) + 1 = 3x - 1
$$

---

### Step 2: Substitute $ x = 4 $

Now that the expression is simplified to $ 3x - 1 $, we substitute $ x = 4 $:

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

---

### 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 $3 \cdot 4 -1 =\boxed{11}$.