Every prime number greater than 3 is of the fom `6k + 1` or `6k + 5`, where k is some integer.

Show that any positive odd integer is of the form 6q + 1 or 6q + 3 or 6q + 5, where q is some integer.

Show that every positive odd integer is of the form 4q + 1 or 4q + 3, where q is some integer.

The probabilitty of getting a number greater than 6 is :

State true or false for the following statements : The random variable X has probability distribution P(X) of the following form, where k is some number: P(X) ={{:( k"," ,if x = 0),(2k"," ,if x =1),(3k ",", if x =2),(0",", "otherwise"):} Then k = 6

The random variable X has a probability distribution P(X) of the following form, where k is some number : P(x)={(k,, if,, x=0),(2k,, if,, x=1),(3k,, if,, x=2),(0,,,,):} Determine the value of k.

The random variable X has a probability distribution P(X) of the following form, where k is some number : P(x)={(k,, if,, x=0),(2k,, if,, x=1),(3k,, if,, x=2),(0,,,,):} find P(X<2), P(Xle2), P(Xge2) .

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.