To find the smallest number by which 27783 should be multiplied to get a perfect cube, we need to factorize 27783 and then adjust its prime factorization so that each exponent is a multiple of 3.

Let's start by factorizing 27783:

1. Check for divisibility by 3:

   • Sum of the digits of 27783: $2 + 7 + 7 + 8 + 3 = 27$
   • Since 27 is divisible by 3, 27783 is also divisible by 3.
   • $27783 \div 3 = 9261$

2. Factorize 9261:

   • Sum of the digits of 9261: $9 + 2 + 6 + 1 = 18$
   • Since 18 is divisible by 3, 9261 is also divisible by 3.
   • $9261 \div 3 = 3087$

3. Factorize 3087:

   • Sum of the digits of 3087: $3 + 0 + 8 + 7 = 18$
   • Since 18 is divisible by 3, 3087 is also divisible by 3.
   • $3087 \div 3 = 1029$

4. Factorize 1029:

   • Sum of the digits of 1029: $1 + 0 + 2 + 9 = 12$
   • Since 12 is divisible by 3, 1029 is also divisible by 3.
   • $1029 \div 3 = 343$

5. Factorize 343:

   • Sum of the digits of 343: $3 + 4 + 3 = 10$
   • 343 is not divisible by 3, so check for divisibility by 7:
     • $343 \div 7 = 49$
     • $49 \div 7 = 7$
     • $7 \div 7 = 1$

So, the prime factorization of 27783 is: $27783 = 3^4 \times 7^3$

To make this a perfect cube, each exponent in the prime factorization must be a multiple of 3.

• For $3^4$: The exponent 4 needs to be adjusted to the nearest multiple of 3, which is 6. So we need 2 more factors of 3.
• For $7^3$: The exponent 3 is already a multiple of 3, so no adjustment is needed.

Therefore, the smallest number by which 27783 should be multiplied to get a perfect cube is: $3^2 = 9$

Thus, the smallest number is $\boxed{9}$.

Would you like more details or have any questions?

Here are 8 related questions:

1. What is the prime factorization method?
2. How do you check for divisibility by different numbers?
3. What is a perfect cube?
4. How do you determine the smallest multiplier to achieve a perfect square?
5. How can you find the smallest number by which a given number should be divided to get a perfect cube?
6. What are the properties of cube numbers?
7. Can you explain the exponent rules for cube numbers?
8. What is the difference between perfect squares and perfect cubes?

Tip: When factorizing a large number, always start with the smallest prime numbers and check divisibility systematically.