R. D. Sharma Solutions for Chapter: Real Numbers, Exercise 1: EXERCISE 1.1

Show that the square of an odd positive integer is of the form $8q+1$, for some integer $q$.

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

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

Show that the cube of a positive integer is of the form $6q+r$, where $q$ is an integer and $r=0, 1, 2, 3, 4, 5$.

Show that one and only one out of $n, n+4, n+8, n+12$ and $n+16$ is divisible by $5$, where $n$ is any positive integer.

Show that the square of an odd positive integer can be of the form $6q+1$ or $6q+3$ for some integer $q$.

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

Show that the square of any positive integer cannot be of the form $3m+2$, where $m$ is a natural number.