Real Numebrs Chapter 1 class 10 Basic Concepts
1. Euclid's division lemma :
Given positive integers $a$ and $b$, there exist whole numbers $q$ and $r$ satisfying $a=b q+r$, $0 \leq r < b$
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 $a=b q+r, 0 \leq r< b$.
Step 2 : If $r=0$, the HCF is $b$. If $r \neq 0$, 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 $\operatorname{HCF}(a, b)$. Also, $\operatorname{HCF}(a, b)=\operatorname{HCF}(b, r)$.
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 $a^{2}$, then $p$ divides $a$, where $a$ is a positive integer.
5. $\sqrt{2}, \sqrt{3}$ are irrationals.
6. Let $x$ be a rational number whose decimal expansion terminates. Then we can express $x$ in the form $\frac{p}{q}$, 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 $x=\frac{p}{q}$ 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 $x=\frac{p}{q}$ 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).
$\mathrm{HCF}(p, q, r) \times \operatorname{LCM}(p, q, r) \neq p \times q \times r$, where $p, q, r$ are positive integers. However, the following results hold good for three numbers $p, q$ and $r:$
$$
\begin{aligned}
& \operatorname{LCM}(p, q, r)=\frac{p \cdot q \cdot r \cdot \mathrm{HCF}(p, q, r)}{\operatorname{HCF}(p, q) \cdot \mathrm{HCF}(q, r) \cdot \operatorname{HCF}(p, r)} \\
& \operatorname{HCF}(p, q, r)=\frac{p \cdot q \cdot r \cdot \operatorname{LCM}(p, q, r)}{\operatorname{LCM}(p, q) \cdot \operatorname{LCM}(q, r) \cdot \operatorname{LCM}(p, r)}
\end{aligned}
$$