Given N is a positive integer less than 31, how many values can n take if (n +…

Key idea: let m = n + 1. We need m to divide (m - 1)!.

If m is prime then m does not divide (m - 1)! because none of its prime factors appear among 1, 2, ..., m - 1.

If m is composite and m ≠ 4, write m = a·b with 1 < a ≤ b < m. Then both a and b occur among 1, 2, ..., m - 1, so their product m divides (m - 1)!.

The single exception is m = 4, because 4 = 2·2 but (m - 1)! = 3! = 6 contains only one factor 2, so 4 does not divide 3!.

Therefore m divides (m - 1)! for every composite m except m = 4.

Here m = n + 1 ranges from 2 to 31 (since n is a positive integer less than 31).

Composite values of m in the interval 2..31 are: 4, 6, 8, 9, 10, 12, 14, 15, 16, 18, 20, 21, 22, 24, 25, 26, 27, 28, 30.

There are 19 such composite m, but exclude the exception m = 4, leaving 18 values of m. Each such m corresponds to n = m - 1, so there are 18 valid values of n.

Answer: 18