Final answer:
The smallest natural number by which 378 should be divided so that the quotient is a perfect cube is 2.
Step-by-step explanation:
To find the smallest natural number by which 378 should be divided so that the quotient is a perfect cube, we need to factorize 378 and look for the smallest power of a prime number that can divide 378 and result in a perfect cube quotient. The prime factorization of 378 is 2 * 3^3 * 7. Since a perfect cube requires each prime factor to have a multiple of 3 as an exponent, we need to divide 378 by the smallest prime factor, which is 2. Therefore, the answer is option A) 2.