Identifying the Perfect Cube Roll Number

A perfect cube is a number that can be obtained by multiplying an integer by itself three times. In mathematical terms, a number '$n$' is a perfect cube if there exists an integer '$k$' such that $n = k^3$.

Analyzing the Options

We need to check which of the given roll numbers is a perfect cube:

• Option 1: 216

  Let's find the cube root of 216. We can test small integers:

  $1^3 = 1$

  $2^3 = 8$

  $3^3 = 27$

  $4^3 = 64$

  $5^3 = 125$

  $6^3 = 6 \times 6 \times 6 = 216$

  Since $216 = 6^3$, 216 is a perfect cube.

• Option 2: 225

  We can check integers around the expected cube root. We know $6^3 = 216$. Let's check $7^3 = 343$. 225 falls between these. Alternatively, we can recognize $225 = 15^2$. It is a perfect square, not a perfect cube.

• Option 3: 243

  We know $6^3 = 216$ and $7^3 = 343$. 243 falls between these. Prime factorization of 243 is $3 \times 3 \times 3 \times 3 \times 3 = 3^5$. Since the exponents are not multiples of 3, it's not a perfect cube.

• Option 4: 256

  We know $6^3 = 216$ and $7^3 = 343$. 256 falls between these. We can recognize $256 = 16^2$. It is a perfect square, not a perfect cube.

Conclusion

Based on the analysis, only the roll number 216 is a perfect cube ($6^3$).