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 the square of any positive integer is of the form 3p or 3p + 1.

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

Show that, if n be any positive integer greater than 1, then (2^(3n) - 7n - 1) is divisible by 49.

If m is an even integer, show that m^(2) is also an even integer.

Prove by induction that the sum of the cubes of three successive positive integers is divisible by 9.

Two pairs of statements p and q are given below . Combine theses two statements using the biconditional phrase "if and only if " . p : If the sum of digits in a positive integer n is divisible by 9 , then the number is divisibel by 9 q : If a positive integer n is divisible by 9 , then the sum of the digits in n is divisible by 9 .

If n (> 1)is a positive integer, then show that 2^(2n)- 3n - 1 is divisible by 9.

An arc of a bridge is semi-elliptical with the major axis horizontal. If the length of the base is 9m and the highest part of the bridge is 3m from the horizontal, then prove that the best approximation of the height of the acr 2 m from the center of the base is 8/3mdot

The number of interger values of m, for which the x coordinate of the point of intersection of the lines y = mx +1 and 3x+4 y =9 is also in integer, is-