# Lecture-2

## Part - 1

In this lecture, I am discussing uses of Euclid’s Division Lemma other than finding HCF or GCD. How to represent Integers in the form of 2q, 2q+1, 3q+1, 3q+2, 4q+1, 4q+2, 4q+3 etc… for Cbse Class 10 Maths Chapter 1 Real Numbers.

## Part - 2

00:00:39 Question 1. 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.

00:08:34 Question 2. Show that any positive odd integer is of the form 4q + 1 or 4q + 3, where q is some integer.

00:16:04 Question 3. Show that any positive odd integer is of the form 6q + 1, or 6q + 3, or 6q + 5, where q is some integer.

## Part - 3

00:03:32 Question 4. Use Euclid’s division lemma to show that the square of any positive integer is either of the form 3m or 3m + 1 for some integer m.

00:20:02 Question 6. Show that the square of any positive integer is either of the form 4q or 4q + 1 for some integer q.

00:28:02 Question 8. Prove that the square of any positive integer is of the form 5q, 5q + 1, 5q + 4 for some integer, q.

00:45:42 Question 5. Show that square of any odd integer is of the form 4m + 1, for some integer m.

00:52:32 Question 7. Show that the square of an odd positive integer is of the form 8m + 1, for some whole number m.

## Part - 4

00:09:40 Question 9. Show that the cube of any positive integer is of the form 4m, 4m + 1 or 4m + 3 for some integer m.

00:01:30 Question 10. Use Euclid’s division lemma to show that the cube of any positive integer is of the form 9m, 9m + 1 or 9m + 8.