Adding Consecutive Odd Numbers:

• The sum of the first few consecutive odd numbers forms a perfect cube:

  

• The number of consecutive odd numbers required to form a cube is always equal to the base number of the cube.

• For example, to obtain , we need 10 consecutive odd numbers.

2. Cubes and Their Prime Factors:

2.1 Prime Factorization of a Number and its Cube:

• Observing the prime factorization of some numbers and their cubes:

  

• Each prime factor of a number appears three times in the prime factorization of its cube.

2.2 Identifying a Perfect Cube:

• A number is a perfect cube if each prime factor in its prime factorization appears three times.

• Example:

  • 216: → Perfect cube ✅

  • 729: → Perfect cube ✅

  • 500: → Not a perfect cube ❌

3. Finding the Smallest Multiple that is a Perfect Cube:

3.1 Understanding the Concept:

• Suppose we have a cuboid with dimensions 15 cm × 30 cm × 15 cm.

• Volume of the cuboid:

  • 15×30×15 = 3×5×2×3×5×3×5

  • 2×3³×5³

• To form a perfect cube, all prime factors must appear in multiples of three.

• The missing factor is 2, so multiplying by 2 will make it a perfect cube.

• The smallest multiple that makes this a perfect cube is 2 × Volume.

Conclusion:

• Perfect cubes follow a pattern in summation and prime factorization.

• Prime factorization helps in identifying whether a number is a perfect cube.

• Multiplication by appropriate factors can make a number a perfect cube.