What is the smallest number by which 1600 must be divided so that the quotient is a perfect cube.

Answer

To find the smallest number by which 1600 must be divided so that the quotient is a perfect cube, we can follow these steps:

### Step 1: Prime Factorization of 1600
First, we need to find the prime factorization of 1600.

- Divide 1600 by 2:
- 1600 ÷ 2 = 800
- Divide 800 by 2:
- 800 ÷ 2 = 400
- Divide 400 by 2:
- 400 ÷ 2 = 200
- Divide 200 by 2:
- 200 ÷ 2 = 100
- Divide 100 by 2:
- 100 ÷ 2 = 50
- Divide 50 by 2:
- 50 ÷ 2 = 25
- Divide 25 by 5:
- 25 ÷ 5 = 5
- Divide 5 by 5:
- 5 ÷ 5 = 1

Thus, the prime factorization of 1600 is:
\[ 1600 = 2^6 \times 5^2 \]

### Step 2: Grouping the Factors
To make the quotient a perfect cube, we need to group the prime factors in sets of three.

- For \(2^6\): We can group it as \( (2^3) \times (2^3) \) which is fine since both groups contain three 2's.
- For \(5^2\): We have only two 5's, which cannot form a complete group of three.

### Step 3: Determine the Number to Divide
Since we need a complete group of three for the factor of 5, we need one more 5 to make it \(5^3\). Therefore, we need to divide by \(5^1\) to achieve this.

### Step 4: Calculate the Smallest Number
The smallest number we need to divide 1600 by is:
\[ 5^1 = 5 \]

### Step 5: Verify the Result
After dividing 1600 by 5, we get:
\[ \frac{1600}{5} = 320 \]
Now, we can check the prime factorization of 320:
- \(320 = 2^6 \times 5^1\)

Now, we can group them:
- \(2^6\) can be grouped as \( (2^3) \times (2^3) \)
- \(5^1\) is still not a complete group, but we can see that we need to divide by \(5^1\) to make it a perfect cube.

### Conclusion
The smallest number by which 1600 must be divided to make the quotient a perfect cube is:
\[ \text{Answer: } 5 \]

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