Find the smallest number by which `8768` must be divided so that the quotient is a perfect cube.

Options

A. `147`
B. `117`
C. `127`
D. `137`

Answer

To find the smallest number by which `8768` must be divided so that the quotient is a perfect cube, we will follow these steps:

### Step 1: Prime Factorization of 8768
First, we need to factorize the number `8768`. Since it is an even number, we can divide it by `2`.

1. \( 8768 \div 2 = 4384 \)
2. \( 4384 \div 2 = 2192 \)
3. \( 2192 \div 2 = 1096 \)
4. \( 1096 \div 2 = 548 \)
5. \( 548 \div 2 = 274 \)
6. \( 274 \div 2 = 137 \)

Now we have \( 137 \) which is a prime number. Therefore, the complete prime factorization of `8768` is:
\[ 8768 = 2^6 \times 137^1 \]

### Step 2: Analyze the Exponents
To determine if a number is a perfect cube, all the exponents in its prime factorization must be multiples of `3`.

- The exponent of `2` is `6`, which is a multiple of `3`.
- The exponent of `137` is `1`, which is **not** a multiple of `3`.

### Step 3: Adjust the Exponents
To make the exponent of `137` a multiple of `3`, we need to increase it to the next multiple of `3`, which is `3`.

To do this, we need to remove \( 137^{1} \) and make it \( 137^{0} \) (which means we need to divide by `137`).

### Step 4: Conclusion
Thus, the smallest number by which `8768` must be divided to make the quotient a perfect cube is:
\[ \text{Required number} = 137 \]

### Final Answer
Therefore, the answer is \( 137 \).

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