Numbers and Sequences — Chapter Summary

For any two positive integers

For any two positive integers a and b, there exist unique integers q and r such that a = bq + r, where 0 ≤ r < b. Here, a is the dividend, b is the di

This is a step

This is a step-by-step method to find the Highest Common Factor (HCF) of two positive integers. The algorithm repeatedly applies the division lemma: d

Every positive integer greater than 1

Every positive integer greater than 1 can be expressed as a product of prime numbers in exactly one way (ignoring the order of factors). For example,

Two integers a and b

Two integers a and b are congruent modulo n (written as a ≡ b (mod n)) if they leave the same remainder when divided by n, or equivalently, if a - b i

If a ≡ b (mod m)

If a ≡ b (mod m) and c ≡ d (mod m), then: (a + c) ≡ (b + d) (mod m); (a - c) ≡ (b - d) (mod m); (a × c) ≡ (b × d) (mod m). These properties allow us t