What is the smallest number by which `8788` must be divided so that the quotient is a perfect cube.

Options

A. 6
B. 8
C. 4
D. 3

Answer

To find the smallest number by which `8788` must be divided so that the quotient is a perfect cube, we will follow these steps:

### Step 1: Prime Factorization of 8788
First, we need to perform the prime factorization of `8788`.

- Divide `8788` by `2` (the smallest prime number):
- \( 8788 \div 2 = 4394 \)
- Divide `4394` by `2` again:
- \( 4394 \div 2 = 2197 \)
- Now, `2197` is not divisible by `2`. We try the next prime number, which is `3`, but `2197` is not divisible by `3`. Next, we try `13`:
- \( 2197 \div 13 = 169 \)
- Finally, we factor `169`:
- \( 169 = 13 \times 13 \)

So, the prime factorization of `8788` is:
\[ 8788 = 2^2 \times 13^3 \]

### Step 2: Determine the Exponents
To form a perfect cube, all the exponents in the prime factorization must be multiples of `3`.

- For `2^2`, the exponent `2` is not a multiple of `3`. To make it a multiple of `3`, we need to increase it to `3`. This means we need one more `2`, which is \( 2^{3-2} = 2^1 \).
- For `13^3`, the exponent `3` is already a multiple of `3`.

### Step 3: Calculate the Smallest Number to Divide
To make the quotient a perfect cube, we need to divide `8788` by the factor we found:
- We need to divide by \( 2^1 = 2 \).

### Step 4: Final Calculation
Now, we can calculate the smallest number by which `8788` must be divided:
- The smallest number is \( 2 \).

### Conclusion
Thus, the smallest number by which `8788` must be divided to make the quotient a perfect cube is **2**.

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