Use division algorithm to show that the square of any positive integer is of the form 3p or 3p + 1.

Use division algorithm to show that the cube of any positive integer is of the form 9 m, 9m + 1 or 9m + 8.

Use division algorithm to 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.

Show that every positive even integer is of the form 2q, and that every positive odd integer is of the form 2q + 1, where q is some integer.

If n is a positive integer then .^(n)P_(n) =

Show that for any positive integer 3^(2n+2)-8n-9 is divisible by 64 .

Prove that the square of an odd integer can be expressed in the form 8p + 1 where p = 0 or a positive integer.

The contrapositives of few conditional statements are given below , find the conditional statements : If the sum of digits in a positive integer n is not divisible by 3 , then the integer n is not divisible by 3 .

If B ,C are square matrices of order na n difA=B+C ,B C=C B ,C^2=O , then without using mathematical induction, show that for any positive integer p ,A^(p+1)=B^p[B+(p+1)C] .

A(3, 0) and B(-3, 0) are two given points and P is a moving point : if bar(AP) = 2bar(BP) for all positions of P, show that the locus of P is a circle. Find the radius of the circle.