Prove that if p is a prime number greater than 2,
then the difference `[(2+sqrt(5))^p]-2^(p+1)` is divisible by p, where [.] denotes greatest integer.

f(x)=1/sqrt([x]^(2)-[x]-6) , where [*] denotes the greatest integer function.

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

The value of lim_(nto oo)(sqrt(n^(2)+n+1)-[sqrt(n^(2)+n+1)]) where [.] denotes the greatest integer function is

The value of lim_(xtoa)[sqrt(2-x)] , where a in[0,1/2] and [.] denotes the greatest integer function is:

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 ge 5 is a prime number, show that p^2 + 2 is a composite number.

The plane denoted by P_1 : 4x+7y+4z+81=0 is rotated through a right angle about its line of intersection with plane P_2 : 5x+3y+10z=25 . If the plane in its new position be denoted by P, and the distance of this plane from the origin is d, then the value of [(k)/(2)] (where[.] represents greatest integer less than or equal to k) is....

Consider a number N = 2 1 P 5 3 Q 4. The number of ordered pairs (P,Q) so that the number 'N' is divisible by 44, is

The point ([p+1], [p]) is lying inside the circle x^2+y^2-2x-15=0 and x^2+y^2-2x-7=0 . Then the set of all values of p is (where [.] represents the greatest integer function)

Let R =(5 sqrt5+11)^(2n+1) and and f=R-[R], where [ ] denotes the greatest integer function, prove that Rf=4^(2n+1) .