The smallest number by which 243000 be divided so that the quotient is a perfect cube is

Options

A. a) 3
B. b) 27
C. c) 9
D. d) 1

Answer

To find the smallest number by which 243,000 must be divided so that the quotient is a perfect cube, we can follow these steps:

### Step 1: Prime Factorization of 243,000
First, we need to factor 243,000 into its prime factors.

243,000 can be expressed as:
\[ 243,000 = 243 \times 1000 \]

### Step 2: Factor Each Component
Now, we will factor each component separately.

- For 243:
\[ 243 = 3^5 \]
(243 is \(3 \times 3 \times 3 \times 3 \times 3\))

- For 1000:
\[ 1000 = 10^3 = (2 \times 5)^3 = 2^3 \times 5^3 \]

### Step 3: Combine the Factors
Now, we combine these factors:
\[ 243,000 = 3^5 \times 2^3 \times 5^3 \]

### Step 4: Analyze the Exponents
To determine if the number is a perfect cube, we need to check the exponents of the prime factors. A number is a perfect cube if all the exponents in its prime factorization are multiples of 3.

- For \(3^5\): The exponent is 5, which is not a multiple of 3. To make it a multiple of 3, we need to reduce it to the nearest lower multiple of 3, which is 3. Thus, we need to remove \(5 - 3 = 2\) from the exponent of 3.
- For \(2^3\): The exponent is already 3, which is a multiple of 3.
- For \(5^3\): The exponent is also 3, which is a multiple of 3.

### Step 5: Determine the Smallest Number to Divide
Since we need to reduce the exponent of \(3\) from \(5\) to \(3\), we need to divide by \(3^2\) (which is \(9\)).

### Conclusion
Thus, the smallest number by which 243,000 must be divided to make the quotient a perfect cube is:
\[ \boxed{9} \]

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