Prove that integral part of `(8+3sqrt(7))^n` is an odd integer.

Integral part of (sqrt2+1)^6 is

Show that the integral part of (5 + 2 sqrt(6))^(n) is odd where n is natural number

Show that the integral part of (5 sqrt(5) + 11)^(2n +1) is even where n in N .

If n is a positive integer and (3sqrt(3)+5)^(2n+1)=l+f where l is an integer annd 0 lt f lt 1 , then

Prove that the product of 2n consecutive negative integers is divisible by (2n)! .

If [x] and (x) are the integral part of x and nearest integer to x then solve (x)[x]=1

State the converses of the following statement. In each case, also decide whether the converse is true or false.:- If n is an even integer, then 2n + 1 is an odd integer.

By giving a counter example, show that the following statement is false. "If n is an odd integer, then n is prime.”

If A is any square matrix, prove that (A^n)' = (A')^n , where n is any positive integer.

Show that (-sqrt(-1))^(4n+3)=i , where n is a positive integer.