What is the samallest number by which 3087 may be multiplied , so that the product is perfect cube?

Answer

To find the smallest number by which 3087 must be multiplied to make it a perfect cube, we will follow these steps:

### Step 1: Prime Factorization of 3087
First, we need to perform the prime factorization of 3087.

- Since 3087 is an odd number, it is not divisible by 2.
- Next, we check for divisibility by 3. The sum of the digits (3 + 0 + 8 + 7 = 18) is divisible by 3. So, we divide:
\[ 3087 \div 3 = 1029 \]
- We check 1029 for divisibility by 3 again (1 + 0 + 2 + 9 = 12):
\[ 1029 \div 3 = 343 \]
- Now, we check 343. The sum of the digits (3 + 4 + 3 = 10) is not divisible by 3. We check for divisibility by 7:
\[ 343 \div 7 = 49 \]
- Finally, we factor 49:
\[ 49 = 7 \times 7 \]

So, the complete prime factorization of 3087 is:
\[ 3087 = 3^2 \times 7^3 \]

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

- For \(3^2\), the exponent is 2, which is not a multiple of 3. We need one more factor of 3 to make it \(3^3\).
- For \(7^3\), the exponent is already a multiple of 3.

### Step 3: Determine the Smallest Number to Multiply
To make the product a perfect cube, we need to multiply by \(3^1\) (or simply 3) to make the exponent of 3 equal to 3.

### Conclusion
Thus, the smallest number by which 3087 must be multiplied to become a perfect cube is:
\[ \boxed{3} \]

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