Find the smallest number by which 26244 should be divided so that the quotient is a perfect cube .

Answer

To find the smallest number by which 26244 should be divided so that the quotient is a perfect cube, we will follow these steps:

### Step 1: Prime Factorization
We start by performing the prime factorization of 26244.

1. Divide by 2:
- 26244 ÷ 2 = 13122
- 13122 ÷ 2 = 6561
2. Now divide by 3:
- 6561 ÷ 3 = 2187
- 2187 ÷ 3 = 729
- 729 ÷ 3 = 243
- 243 ÷ 3 = 81
- 81 ÷ 3 = 27
- 27 ÷ 3 = 9
- 9 ÷ 3 = 3
- 3 ÷ 3 = 1

Thus, the prime factorization of 26244 is:
\[ 26244 = 2^2 \times 3^8 \]

### Step 2: Analyze the Exponents
Next, we analyze the exponents of the prime factors:
- The exponent of 2 is 2 (which is not a multiple of 3).
- The exponent of 3 is 8 (which is a multiple of 3).

### Step 3: Make Exponents Multiples of 3
To make the quotient a perfect cube, all the exponents in the prime factorization must be multiples of 3.

- For \(2^2\): The nearest multiple of 3 is 3. We need to increase the exponent from 2 to 3, which means we need one more factor of 2.
- For \(3^8\): The exponent is already a multiple of 3 (8 is greater than 6). Thus, no additional factors of 3 are needed.

### Step 4: Calculate the Smallest Number to Divide
To achieve the required exponent for 2, we need to divide by \(2^1\) (since we need one more factor of 2).

Thus, the smallest number to divide by is:
\[ 2^1 = 2 \]

### Step 5: Final Calculation
Now, we need to check if dividing 26244 by 2 gives us a perfect cube:
\[ \frac{26244}{2} = 13122 \]

We can factor 13122:
\[ 13122 = 2^1 \times 3^8 \]
Now, the exponents are:
- For 2: 1 (not a multiple of 3)
- For 3: 8 (is a multiple of 3)

To make it a perfect cube, we need to divide by \(2^1\) again, which means:
\[ 2 \times 2 = 4 \]

Thus, the smallest number by which 26244 should be divided to get a perfect cube is:
\[ 4 \]

### Conclusion
The smallest number by which 26244 should be divided so that the quotient is a perfect cube is **4**.

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