Understanding Perfect Cubes
A perfect cube is a number that can be expressed as the cube of an integer. To determine what number must be multiplied with 5400 to make it a perfect cube, we first need to analyze its prime factorization.
Prime Factorization of 5400
- Start by breaking down 5400 into its prime factors:
- 5400 = 54 × 100
- 54 = 2 × 27 = 2 × 3^3
- 100 = 10 × 10 = (2 × 5) × (2 × 5) = 2^2 × 5^2
- Combining these, we get:
- 5400 = 2^3 × 3^3 × 5^2
Analyzing the Exponents
- The prime factorization shows the exponents:
- 2^3 (exponent is 3)
- 3^3 (exponent is 3)
- 5^2 (exponent is 2)
To be a perfect cube, all the exponents must be multiples of 3.
Making Adjustments
- For 2^3, the exponent 3 is already a multiple of 3.
- For 3^3, the exponent 3 is also a multiple of 3.
- For 5^2, the exponent 2 is not a multiple of 3. We need to increase it to the next multiple of 3, which is 3.
Calculating the Required Multiplier
- To change 5^2 to 5^3, we must multiply by 5^(3-2) = 5^1 = 5.
Conclusion
Thus, the smallest number that must be multiplied with 5400 to make it a perfect cube is 5.

Simply 5400=2*2*2*3*3*3*5*5
which means 2^3*3^3*5^2
so here 5 has less times so least number is 5