TOPICS

Formulae

Real Numbers

1. Given positive integers a and b, there exist unique integers q and r satisfying a = bq + r, 0 ≤ r < b.< b 2. If p is a prime and p divides a², then p divides a, where a is a positive integer.

1. Let x be a rational number whose decimal expansion terminates. Then we can express x in the form 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. 2. Let x = p/q be a rational number, such that the prime factorisation of q is of the form 2^{m}× 5^{n}, where n, m are non-negative integers. Then x has a decimal expansion which terminates. 3. Let x = p/q be a rational number, such that the prime factorisation of q is not of the form 2^{m}× 5^{n}, where n, m are non-negative integers. Then x has a decimal expansion which is non-terminating repeating (recurring).

1. for any two positive integers a and b, HCF(a,b) × LCM(a,b) = a × b. 2. for any two positive integers a and b, HCF(a,b) = (a × b) × LCM(a,b). 3. for any two positive integers a and b, LCM(a,b) = (a × b) x HCF(a,b). 4. HCF(p,q,r) × LCM(p,q,r) ≠ p × q × r, where p, q, r are positive integers. However, the following results hold good for three numbers p, q and r : i) LCM(p,q,r) = [p.q.r.HCF(p,q,r)] ÷ [HCF(p,q).HCF(q,r).HCF(p,r)] ii) HCM(p,q,r) = [p.q.r.LCM(p,q,r)] ÷ [LCM(p,q).LCM(q,r).LCM(p,r)]

CLASSES