**QUESTION:-**The primorial function for a natural number N (denoted by N#) is defined as the product of all the prime numbers less than or equal to N. 0# is defined to be equal to 1.

If N# = (N – 1)#, which of the following is necessarily false?

**OPTIONS:-**

1) N is a composite number and (N − 1) is a prime number

2) N and (N – 1) are both composite numbers

3) N is a perfect square and (N – 1) is a perfect cube

4) N is a prime number

5) More than 1 of above

**Solution**We have 4# = 3 × 2 and 3# = 3 × 2, so option 1 can be true.

Similarly, we have 8# = 9# = 7 × 5 × 3 × 2, so options 2 and 3 can also be true.

Now, if N is a prime number, N# will include N in its product.

However, (N – 1)# will not include N in its product, all other terms in the product being same.

N# = (N – 1)# in not possible in that case.

Hence, option 4.

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