Key Points

Every composite number can be expressed (factorised) as a product of prime numbers, and this factorization is unique, apart from the order in which the prime factors occur.

The Highest Common Factor (HCF) is the product of the smallest power of each common prime factor in the numbers. For example, if $a = 2^3 \times 3^2$ and $b = 2^2 \times 3^3$, then $\text{HCF}(a, b) = 2^2 \times 3^2$.

The Least Common Multiple (LCM) is the product of the greatest power of each prime factor involved in the numbers. For example, if $a = 2^3 \times 3^2$ and $b = 2^2 \times 3^3$, then $\text{LCM}(a, b) = 2^3 \times 3^3$.

For any two positive integers $a$ and $b$, the product of their HCF and LCM is equal to the product of the two numbers. The formula is $\text{HCF}(a, b) \times \text{LCM}(a, b) = a \times b$.

The product rule for two numbers does not apply to three numbers. In general, for three positive integers $p, q, r$, it is not true that $\text{HCF}(p, q, r) \times \text{LCM}(p, q, r) = p \times q \times r$.

A number ends with the digit 0 only if its prime factorization contains both 2 and 5. For any natural number $n$, $6^n = (2 \times 3)^n$ can never end in 0 as it lacks the prime factor 5.

A number is composite if it has factors other than 1 and itself. An expression like $7 \times 11 \times 13 + 13$ is composite because it can be factored into $13 \times (7 \times 11 + 1)$, which is $13 \times 78$.

A key theorem for proving irrationality states that if a prime number $p$ divides the square of a positive integer $a$ (i.e., $p$ divides $a^2$), then $p$ must also divide $a$.

To prove that a number like $\sqrt{3}$ is irrational, we assume it is rational, so $\sqrt{3} = \frac{a}{b}$ where $a$ and $b$ are coprime. This assumption leads to a contradiction that both $a$ and $b$ are divisible by 3, proving the initial assumption was false.

The sum or difference of a rational number and an irrational number is always irrational. The product or quotient of a non-zero rational number and an irrational number is also always irrational.