M L Aggarwal Solutions for Chapter: Real Numbers, Exercise 1: Exercise 1.1

If $n$ is a positive odd integer, then show that $n^2-1$ is divisible by $8$

Show that the square of any positive integer cannot be of the form $6m+2 or 6m+5$ for any integer $m$

Prove that one and only one out of any three consecutive positive integers is divisible by $3$.

The values of the remainder $r$, when a positive integer $n$ is divided by $3 are 0 and 1$ only. Justify your answer.

Whether every positive integer can be of the form $4q+2$, where $q$ is an integer. Justify your answer.

"The product of two consecutive positive integers is divisible by $2$". Justify your answer.

"The product of three consecutive positive integers is divisible by $6$". Justify your answer.

A positive integer is of the form $3q+1$, $q$ being a natural number. Can you write its square in any form other than $3m+1 i.e. 3m \text{or} 3m+2$ for some integer $m$? Justify your answer.