Real Numbers & Fundamental Theorem of Arithmetic

Every composite number can be factorized uniquely into prime factors apart from order.

$\text{LCM} \times \text{HCF} = a \times b \Rightarrow \text{LCM} \times 12 = 1800 \Rightarrow \text{LCM} = 150$.

The square root of any non-square prime number is irrational.

$125 = 5^3$. Maximum exponent of 2 or 5 in denominator is 3, so decimal expansion terminates after 3 places.

$144 = 16 \times 9 = 2^4 \times 3^2$. Exponent of 2 is 4.

Smallest prime = 2, smallest composite = 4. $\text{HCF}(2, 4) = 2$.

Co-prime numbers have no common prime factors except 1.

Product of non-zero rational and irrational is always irrational.

$196 = 14^2 = 2^2 \times 7^2$. Sum of exponents = $2 + 2 = 4$.

$96 = 2^5 \times 3$, $404 = 2^2 \times 101$. $\text{HCF} = 2^2 = 4$.