Is 1188 a perfect cube?
To determine whether 1188 is a perfect cube or not, we need to check if it can be expressed as the cube of a natural number. In other words, we need to find a natural number whose cube is equal to 1188.

Finding the smallest natural number whose cube is 1188:
To find the smallest natural number whose cube is 1188, we can use the prime factorization method. We factorize 1188 into its prime factors and check if any of the factors occur three times.

The prime factorization of 1188 can be calculated as follows:
1188 ÷ 2 = 594
594 ÷ 2 = 297
297 ÷ 3 = 99
99 ÷ 3 = 33
33 ÷ 3 = 11

So, the prime factorization of 1188 is 2 × 2 × 3 × 3 × 11.

Now, we check if any of the prime factors occur three times. If they do, then the number is a perfect cube. In this case, we can see that none of the prime factors occur three times. Therefore, 1188 is not a perfect cube.

Dividing 1188 to obtain a perfect cube:
To obtain a perfect cube from 1188, we need to divide it by a suitable number so that the quotient is a perfect cube. Let's find the smallest such number.

Since the prime factorization of 1188 is 2 × 2 × 3 × 3 × 11, we need to find a number that will divide 1188 and result in a quotient that has only the prime factors occurring in groups of three.

The smallest number that satisfies this condition is 2 × 2 × 3 × 3 = 36, which is a perfect cube (6^3).

Therefore, to obtain a perfect cube from 1188, it should be divided by 36. The quotient would then be a perfect cube.

Summary:
- 1188 is not a perfect cube.
- The smallest natural number that can divide 1188 to obtain a perfect cube is 36.
- Dividing 1188 by 36 will result in a quotient that is a perfect cube.