In this chapter, you have studied the following points:

1.  Euclid’s division lemma :

Given positive integers a and b, there exist whole numbers q and r satisfying a = bq + r,
0 ≤ r < b.

2.  Euclid’s division algorithm : This is based on Euclid’s division lemma. 
According to this, the HCF of any  two positive integers a and b, with a > b, is obtained as follows:
Step 1 : Apply the division lemma to find q and r where a = bq + r, 0 ≤ r < b.

Step 2 : If r = 0, the HCF is b. If r ≠ 0, apply Euclid’s lemma to b and r.

Step 3 : Continue the process till the remainder is zero. The divisor at this stage will be
HCF (a, b). Also, HCF(a, b) = HCF(b, r).

3.  The Fundamental Theorem of Arithmetic :
Every composite number can be expressed (factorised) as a product of primes, and this factorisation
is unique, apart from the order in which the prime factors occur.

4.   If p is a prime and p divides a2, then p divides a, where a is a positive integer.

5.   To prove that √2, √3  are irrationals.

6.  Let x be a rational number whose decimal expansion terminates. Then we can express x in the 
form   ${p}/{q}$ , where ${p}$ and ${q}$ are coprime, 
and the prime factorisation of ${q}$ is of the form ${2^n}{5^m}$, where ${n}$,${m}$ are non-negative integers. 

7.  Let x = ${p}/{q}$ be a rational number, such that the prime factorisation of q is of the form ${2^n}{5^m}$, 
where n, m are non-negative integers. Then x has a decimal expansion which terminates.

8.  Let x = ${p}/{q}$  be a rational number, such that the prime factorisation of $q$ is not of the form ${2^n}{5^m}$,
where ${n}$, ${m}$,  are non-negative integers. 
Then$ $x has a decimal expansion which is non-terminating repeating (recurring).