If *X* is the correct number that should replace the wrong number in the series $229, 256, 320, 441, 661, 1004$, then which of the following statements is/are correct?

I. *X* is completely divisible by $15$.

II. (*X* + $2$) is a prime number.

Question

If *X* is the correct number that should replace the wrong number in the series $229, 256, 320, 441, 661, 1004$, then which of the following statements is/are correct?

I. *X* is completely divisible by $15$.

II. (*X* + $2$) is a prime number.

Accepted Answer

**Solution:**

1. Analyze the given series: $229, 256, 320, 441, 661, 1004$.

2. Check the differences between consecutive terms:
   - $256 - 229 = 27$
   - $320 - 256 = 64$
   - $441 - 320 = 121$
   - $661 - 441 = 220$
   - $1004 - 661 = 343$

3. Notice that $27 = 3^3$, $64 = 4^3$, $121 = 11^2$, $220$ is not a perfect cube or square, and $343 = 7^3$.

4. The difference $220$ seems out of place. Let's assume $X$ replaces $661$ to make the pattern consistent.

5. The differences appear to be cubes of consecutive integers:
   - $27 = 3^3$
   - $64 = 4^3$
   - $125 = 5^3$
   - $216 = 6^3$
   - $343 = 7^3$

6. So, the differences should be $27, 64, 125, 216, 343$.

7. Calculate the correct term replacing $661$:
   - Starting from $229$:
     - $229 + 27 = 256$
     - $256 + 64 = 320$
     - $320 + 125 = 445$
     - $445 + 216 = 661$
     - $661 + 343 = 1004$

8. Therefore, the correct number $X$ replacing $441$ is $445$.

9. Check statement I: Is $445$ divisible by $15$?
   - $15 	imes 29 = 435$
   - $15 	imes 30 = 450$
   - So, $445$ is not divisible by $15$.

10. Check statement II: Is $X + 2 = 445 + 2 = 447$ a prime number?
    - $447$ is divisible by $3$ (since $4 + 4 + 7 = 15$, divisible by $3$), so not prime.

11. Both statements I and II are incorrect.

**Answer:** Both I and II incorrect

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