Which of the following statements is true?

Explanation:

For this type of problems, just try to discard given options by taking suitable counter examples..

For option (1). Let n = 16 then

(n - 1)! + 3 = 15! + 3 = even + odd = odd

⇒ No any even integer divide it ie to (n - 1)! + 3

⇒ option (1) is false. (Because z not true for all)

For option (2). taken n = 17, then

∵ (n - 1)! = 16! , Here n = 17 is prime so it will never divide 16!

⇒ option (2) false.

For option (4). take n = 16, then n² = 16² + n! + 1 = 16 ! + 1 Here 16² is even while 16! + 1 is odd integer So 16² + 16! + 1 

⇒ option (4) is false.

For option (3) is true [consider any n ≥ 16 even]

Note: Proof for this option is too long, so just pay to understand with example.

\(=\frac{p_1^{r_1} p_2^{r_2} \cdots p_n^{r_n}}{p_1^{r_1-1} \cdot\left(p_1-1\right) \cdot p_2^{r_2-1}\left(p_2-1\right) \ldots p_n^{r_{n-1}}\left(p_{n-1}\right)}\)

\(=\frac{p_1 p_2 \cdots p_n}{\left(p_1-1\right)\left(p_2-1\right) \cdots\left(p_n-1\right)}\) = integer (given) = p/n¹/n

∵ (p₁ - 1) × p₁ ⇒ ∃ some other prime p₂ S.t (p₁ - 1)|p₂

But ∵  p₂ is also a prime, so not divisible by any of integer except 1 .

(there of one prime factor. is 2, then n tar as at most two distinct prime faster else one.

thus, option (3) is true