1. The problem asks us to find the smallest number by which 243 must be multiplied to become a perfect cube. 2. First, factorize 243 into its prime factors. Since 243 is $3^5$, we have $$243 = 3^5.$$ 3. A perfect cube requires each prime factor to have an exponent that is a multiple of 3. 4. Currently, 243 has $3^5$, and 5 is not a multiple of 3. The next multiple of 3 after 5 is 6. 5. To make the exponent of 3 equal to 6, multiply by $3^{6-5} = 3^1 = 3$. 6. Therefore, the smallest number to multiply 243 by to get a perfect cube is 3. 7. The perfect cube obtained is $$243 \times 3 = 729 = 3^6 = (3^2)^3 = 9^3.$$