Find the smallest number by which 243 must be multiplied to obtain a perfect cube.

To determine the smallest number by which 243 must be multiplied to form a perfect cube, we first need to factor 243 into its prime components.

Prime Factorization of 243

243 can be expressed as:

• 243 = 3 × 3 × 3 × 3 × 3 = 3⁵

Understanding Perfect Cubes

A perfect cube is a number that can be expressed as the cube of an integer. For a number to be a perfect cube, each prime factor's exponent must be a multiple of 3.

Adjusting the Exponents

In the case of 243, the exponent of 3 is 5. To make this a multiple of 3, we need to adjust it:

• The nearest multiple of 3 greater than 5 is 6.
• To reach 6, we need to add 1 to the exponent of 3.

Finding the Required Multiplier

To achieve this, we multiply by 3¹ (which is 3) to increase the exponent from 5 to 6.

Final Answer

Thus, the smallest number by which 243 must be multiplied to obtain a perfect cube is 3.