Answer :
Answer:
Step-by-step explanation:
Sure, let's start by selecting two pairs of composite numbers between 5 and 15:
Pair 1: 6 and 8
Pair 2: 9 and 10
(i) **Finding H.C.F. and L.C.M. of each pair:**
**Pair 1 (6 and 8):**
H.C.F. of 6 and 8:
Factors of 6: 1, 2, 3, 6
Factors of 8: 1, 2, 4, 8
Common factors: 1, 2
Highest common factor (H.C.F.): 2
L.C.M. of 6 and 8:
Multiples of 6: 6, 12, 18, 24, ...
Multiples of 8: 8, 16, 24, ...
Common multiples: 24
Least common multiple (L.C.M.): 24
**Pair 2 (9 and 10):**
H.C.F. of 9 and 10:
Factors of 9: 1, 3, 9
Factors of 10: 1, 2, 5, 10
Common factors: 1
Highest common factor (H.C.F.): 1
L.C.M. of 9 and 10:
Multiples of 9: 9, 18, 27, ...
Multiples of 10: 10, 20, 30, ...
Common multiples: None until infinity
Least common multiple (L.C.M.): Infinity (as there's no finite L.C.M.)
(ii) **Finding the product of H.C.F. and L.C.M. and the product of each pair of numbers:**
**Pair 1 (6 and 8):**
Product of 6 and 8: \( 6 \times 8 = 48 \)
Product of H.C.F. and L.C.M.: \( 2 \times 24 = 48 \)
**Pair 2 (9 and 10):**
Product of 9 and 10: \( 9 \times 10 = 90 \)
Product of H.C.F. and L.C.M.: \( 1 \times \infty = \infty \)
**Discussion:**
In both pairs, we observed an interesting fact: the product of the H.C.F. and L.C.M. equals the product of the given numbers.
For Pair 1:
Product of 6 and 8 = 48
Product of H.C.F. (2) and L.C.M. (24) = 48
For Pair 2:
Product of 9 and 10 = 90
Product of H.C.F. (1) and L.C.M. (infinity) is not finite.
This observation leads us to a fundamental theorem in number theory called "The Fundamental Theorem of Arithmetic" which states that every integer greater than 1 either is a prime number itself or can be factorized into prime numbers and this factorization is unique up to the order of the factors.
In simpler terms, any composite number can be expressed as a unique product of prime numbers. When we consider the H.C.F. and L.C.M. of two numbers, we're effectively considering the common and unique factors they share (H.C.F.) and all the factors they possess (L.C.M.).
So, regardless of the specific numbers chosen, the product of the H.C.F. and L.C.M. will always be equal to the product of the original numbers due to the unique prime factorization of integers.
