Statement 1: If `p` is a prime number `(p!=2),` then `[(2+sqrt(5))^p]-2^(p+1)` is always divisible by `p(w h e r e[dot]` denotes the greatest integer function). Statement 2: if `n` prime, then `^n C_1,^n C_2,^n C_2 ,^n C_(n-1)` must be divisible by `ndot`

Let R=(5sqrt(5)+11)^(2n+1)a n df=R-[R]w h e r e[] denotes the greatest integer function, prove that Rf=4^(2n+1)

Let R=(5sqrt(5)+11)^(2n+1)a n df=R-[R]w h e r e[] denotes the greatest integer function, prove that Rf=4^(2n+1)

If p=(8+3sqrt(7))^n a n df=p-[p],w h e r e[dot] denotes the greatest integer function, then the value of p(1-f) is equal to a. 1 b. 2 c. 2^n d. 2^(2n)

If p=(8+3sqrt(7))^n a n df=p-[p],w h e r e[dot] denotes the greatest integer function, then the value of p(1-f) is equal to a.1 b. 2 c. 2^n d. 2^(2n)

If p=(8+3sqrt(7))^n a n df=p-[p],w h e r e[dot] denotes the greatest integer function, then the value of p(1-f) is equal to 1 b. 2 c. 2^n d. 2^(2n)

If p=(8+3sqrt(7))^n a n df=p-[p],w h e r e[dot] denotes the greatest integer function, then the value of p(1-f) is equal to a. 1 b. 2 c. 2^n d. 2^(2n)

If p is a natural number, then prove that p^(n+1) + (p+1)^(2n-1) is divisible by p^(2) + p +1 for every positive integer n.

If p is a natural number, then prove that p^(n+1) + (p+1)^(2n-1) is divisible by p^(2) + p +1 for every positive integer n.

Statement-1: the highest power of 3 in .^(50)C_(10) is 4. Statement-2: If p is any prime number, then power of p in n! is equal to [n/p]+[n/p^(2)]+[n/p^(3)] + . . ., where [*] denotes the greatest integer function. a. Statement-1 is true, statement-2 is true, statement-2 is a correct explanation for statement-1 b. Statement-1 is true, statement-2 is true, statement-2 is not a correct explanation for statement-1 c. Statement-1 is true, statement-2 is false d. Statement-1 is false, statement-2 is true

If p be a natural number, then prove that, p^(n+1)+(p+1)^(2n-1) is divisible by (p^(2)+p+1) for every positive integer n.