What is the least number by which 13720 must be divided so that the quotient is a perfect cube?

Options

A. 2
B. 3
C. 5
D. 6

Answer

To find the least number by which 13720 must be divided so that the quotient is a perfect cube, we will follow these steps:

### Step 1: Prime Factorization of 13720
We start by finding the prime factorization of 13720.

1. Divide by 2:
- 13720 ÷ 2 = 6860
- 6860 ÷ 2 = 3430
- 3430 ÷ 2 = 1715 (no longer divisible by 2)

2. Now divide by 5:
- 1715 ÷ 5 = 343

3. Finally, divide by 7:
- 343 ÷ 7 = 49
- 49 ÷ 7 = 7
- 7 ÷ 7 = 1

Thus, the prime factorization of 13720 is:
\[ 13720 = 2^3 \times 5^1 \times 7^3 \]

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

- For \(2^3\): The exponent is 3 (which is already a multiple of 3).
- For \(5^1\): The exponent is 1 (not a multiple of 3).
- For \(7^3\): The exponent is 3 (which is already a multiple of 3).

### Step 3: Determine the Missing Factors
To make the exponent of 5 a multiple of 3, we need to remove the factor of \(5^1\).

### Step 4: Conclusion
The least number by which 13720 must be divided to make the quotient a perfect cube is 5.

### Final Answer
Thus, the answer is:
\[ \text{Required number} = 5 \]

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