For any two positive integers a and b, there exist unique integers q (quotient) and r (remainder) such that:
This is called Euclid's Division Lemma. It is used to find the HCF (Highest Common Factor) of two numbers using the Euclid's Division Algorithm.
Step 1: Apply Euclid's Division Lemma to a and b to find q and r such that a = bq + r.
Step 2: If r = 0, then HCF(a, b) = b. If r ≠ 0, apply the lemma to b and r.
Step 3: Continue until the remainder is 0. The last non-zero remainder is the HCF.
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.
A number is irrational if it cannot be expressed in the form p/q, where p and q are integers and q ≠ 0. The decimal expansion of an irrational number is non-terminating and non-repeating.
Proof that √2 is irrational: Assume √2 = p/q (in lowest terms). Then 2 = p²/q², so p² = 2q². This means p² is even, so p is even. Let p = 2m. Then 4m² = 2q², so q² = 2m², meaning q is also even. But this contradicts our assumption that p/q is in lowest terms. Therefore √2 is irrational.