Find the laest number by which 1323 must be multiplied so that the product is a perfect cube.

Answer

To find the least number by which 1323 must be multiplied so that the product is a perfect cube, we can follow these steps:

### Step 1: Factorize 1323
We start by factorizing the number 1323 into its prime factors.

- Since 1323 is an odd number, it is not divisible by 2.
- We check for divisibility by 3. The sum of the digits of 1323 (1 + 3 + 2 + 3 = 9) is divisible by 3, so we divide:
\[ 1323 \div 3 = 441 \]

- Now, we factor 441. Again, the sum of the digits (4 + 4 + 1 = 9) is divisible by 3, so we divide:
\[ 441 \div 3 = 147 \]

- Next, we factor 147. The sum of the digits (1 + 4 + 7 = 12) is also divisible by 3, so we divide:
\[ 147 \div 3 = 49 \]

- Finally, we factor 49, which is \(7 \times 7\) or \(7^2\).

Putting it all together, we have:
\[ 1323 = 3^3 \times 7^2 \]

### Step 2: Identify the exponents
In the prime factorization \(3^3 \times 7^2\), we observe the exponents:
- The exponent of 3 is 3 (which is already a perfect cube).
- The exponent of 7 is 2 (which is not a perfect cube).

### Step 3: Determine the required factor
For a number to be a perfect cube, all the exponents in its prime factorization must be multiples of 3.

- The exponent of 3 is already a multiple of 3.
- The exponent of 7 is 2. To make it a multiple of 3, we need to increase it to 3.

To do this, we need to multiply by \(7^{3-2} = 7^1 = 7\).

### Step 4: Conclusion
Thus, the least number by which 1323 must be multiplied to make it a perfect cube is:
\[ \boxed{7} \]