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

Question

Accepted Answer

To determine the smallest number by which 8788 must be divided so that the quotient is a perfect cube, we need to analyze the prime factorization of 8788 and see how it can be adjusted to form a perfect cube.

1. **Prime Factorization of 8788:**
   - First, 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. Check divisibility by 3, 5, and 7. It is not divisible by any of these.
   - Check divisibility by 11, 13, and 17. It is not divisible by these either.
   - Check divisibility by 19:
     $$
     2197 \div 19 = 115.63 \quad (\text{not an integer})
     $$
   - Check divisibility by 23:
     $$
     2197 \div 23 = 95.52 \quad (\text{not an integer})
     $$
   - Check divisibility by 29:
     $$
     2197 \div 29 = 75.76 \quad (\text{not an integer})
     $$
   - Check divisibility by 31:
     $$
     2197 \div 31 = 71 \quad (\text{exact division})
     $$
   - Therefore, 2197 is $31 \times 31 \times 31$ or $31^3$.

The complete prime factorization of 8788 is $2^2 \times 31^3$.

2. **Adjusting to a Perfect Cube:**
   - For a number to be a perfect cube, all the exponents in its prime factorization must be multiples of 3.
   - In $2^2 \times 31^3$, the exponent of 2 is 2, which is not a multiple of 3.
   - To make it a perfect cube, we need to divide by $2^2$ (which is 4) to remove the factor of 2 completely.

3. **Verification:**
   - Dividing 8788 by 4 gives:
     $$
     8788 \div 4 = 2197
     $$
   - 2197 is $31^3$, which is a perfect cube.

4. **Explanation of Incorrect Options:**
   - **Option 1 (3):** Dividing 8788 by 3 does not result in a perfect cube because the factor of 2 remains.
   - **Option 3 (5):** Dividing by 5 does not address the factor of 2, and the quotient is not a perfect cube.
   - **Option 4 (6):** Dividing by 6 is equivalent to dividing by 2 and 3, which still leaves a factor of 2 in the quotient, preventing it from being a perfect cube.

Therefore, the smallest number by which 8788 must be divided to result in a perfect cube is indeed 4, making Option 2 the correct answer.