Find the smallest number that should be multiplied with 54000 to make it a…

Solution:

Step 1: Prime-factorise 54000.

54000 = 54 × 1000 = (2 × 3^3) × (2^3 × 5^3) = 2^4 × 3^3 × 5^3.

Step 2: For a number to be a perfect cube, every prime exponent must be a multiple of 3.

• Exponent of 2 is 4. The next multiple of 3 ≥ 4 is 6, so we need 2 more factors of 2 (multiply by 2^2 = 4).

• Exponent of 3 is 3 (already a multiple of 3), so no extra 3 is needed.

• Exponent of 5 is 3 (already a multiple of 3), so no extra 5 is needed.

Step 3: Multiply by 2^2 = 4. This gives 54000 × 4 = 216000 = 60^3, which is a perfect cube.

Answer: 4