| If \[X=28+(1\times 2\times \times 3\times 4\times ...\times 16\times 28)\] and \[Y=17+(1\times 2\times 3\times ....\times 17),\]then which of the following is/are true? |
| (i) X is a composite number |
| (ii) Y is a prime number |
| (iii) \[X-Y\]is a prime number |
| (iv) \[X+Y\] is a composite number |
A) Both (i) and (iv)
B) Both (ii) and (iii)
C) Both (ii) and (iv)
D) Both (i) and (ii)
Correct Answer: A
Solution :
| Sol. [a] We have, \[X=28+(1\times 2\times 3\times ....\times 16\times 28)\] |
| \[X=28+[1+(1\times 2\times 3\times ....\times 16)]\] |
| Hence. X is a composite number. |
| Also, we have |
| \[Y=17+(1\times 2\times 3\times .....\times 17)\] |
| \[=17[1+(1\times 2\times 3\times ...\times 16)]\] |
| Hence, Y is a composite number. |
| Now, \[X-Y=[1+(1\times 2\times ....\times 16)]\,\,(28-17)\] |
| \[[1+(1\times 2\times 3.....\times 16)]\,\,(45)\] |
| \[=[1+(1\times 2\times ...\times 16)]\,\,\,\,(11)\] |
| Hence, \[X-Y\] is a composite number. |
| and \[X+Y=[1+(1\times 2\times 3\times .....\times 16)]\,\,\,(28+17)\] |
| \[=[1+(1\times 2\times 3\times ....\times 16)\times 45]\] |
| Hence, \[X+Y\] is a composite number. |
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