This session covers the concepts like 'Fundamental Theorem of Arithmetic and Relationaship between HCF and LCM.
- Fundamental Theorem of Arithmetic - Prime factorization method, HCF and LCM
- Relationship between HCF and LCM, Word problems on HCF and LCM
Let us recall the concepts in Real Numbers:
1. HCF: Product of the smallest power of each common prime factor in the numbers.
2. LCM: Product of the greatest power of each prime factors involved in the numbers.
3. For any two positive integers \(a\) and \(b\), \(HCF(a,b) \times LCM(a,b) = a \times b.\)
4. Fundamental theorem of Arithmetic: 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 occurs.
5. If \(p\) is a prime and \(p\) divides \(a^2\), then \(p\) divides \(a\), where \(a\) is a positive integer.
6. \(HCF ( p, q, r) \times LCM (p, q, r) \neq p \times q \times r \), where \(p, q, r\) are positive integers,
\(LCM (p, q, r) = \frac {p\times q \times r\times \text{HCF}(p,q, r)}{\text{HCF}(p,q) \times \text{HCF}(q,r) \times \text{HCF}(p,r)}\)
\(HCF (p, q, r) = \frac {p\times q \times r\times \text{LCM}(p,q, r)}{\text{LCM}(p,q)\times \text{LCM}(q,r) \times \text{LCM}(p,r)}\)
\(HCF (p, q, r) = \frac {p\times q \times r\times \text{LCM}(p,q, r)}{\text{LCM}(p,q)\times \text{LCM}(q,r) \times \text{LCM}(p,r)}\)