The relationship between HCF and LCM of two natural numbers isa)HCF×LCM = Product of two natural numbersb)HCF×LCM = Sum of two natural numbersc)HCF×LCM = Difference of two natural numbersd)None of theseCorrect answer is option 'A'. Can you explain this answer?

Question

The relationship between HCF and LCM of two natural numbers isa)HCF&#215;LCM = Product of two natural numbersb)HCF&#215;LCM = Sum of two natural numbersc)HCF&#215;LCM = Difference of two natural numbersd)None of theseCorrect answer is option 'A'. Can you explain this answer?

Answer

The product of LCM and HCF of any two given natural numbers is equivalent to the product of the given numbers.

LCM × HCF = Product of the Numbers

Suppose A and B are two numbers, then.

LCM (A & B) × HCF (A & B) = A × B

Example: Prove that: LCM (9 & 12) × HCF (9 & 12) = Product of 9 and 12.
->LCM and HCF of 9 and 12:
9 = 3 × 3 = 3²
12 = 2 × 2 × 3 = 2² × 3
LCM of 9 and 12 = 2² × 3² = 4 × 9 = 36
HCF of 9 and 12 = 3
LCM (9 & 12) × HCF (9 & 12) = 36 × 3 = 108
Product of 9 and 12 = 9 × 12 = 108
Hence, LCM (9 & 12) × HCF (9 & 12) = 108 = 9 × 12

Answer

Let us assume 2 and 3....l.c.m is 6 and h.c.f is 1 h.c.f×l.c.m= product of numbers 1×6=2×3 6=6

Answer

A  options is the answer

Answer

It's an axiom and cannot be proved you can simply take two nos. and find their product and their hcf and lcm product

Answer

Answer is (a)

Answer

Relation between HCF and LCM
The Highest Common Factor (HCF) and Lowest Common Multiple (LCM) are two important concepts in number theory. They have a relationship that can be expressed mathematically as HCFLCM = Product of two natural numbers. Let's understand this relationship in detail.

Defining HCF and LCM
1. Highest Common Factor (HCF):
- HCF of two or more numbers is the largest positive integer that divides each of the numbers without leaving a remainder.
- For example, the HCF of 12 and 18 is 6 because it is the largest number that divides both 12 and 18 without leaving a remainder.

2. Lowest Common Multiple (LCM):
- LCM of two or more numbers is the smallest multiple that is divisible by each of the numbers.
- For example, the LCM of 12 and 18 is 36 because it is the smallest number that is divisible by both 12 and 18.

Understanding the relationship
The relationship between HCF and LCM can be understood by considering the prime factorization of the numbers involved. Prime factorization is expressing a number as the product of its prime factors.

Let's take two natural numbers, a and b, and express them in terms of their prime factors:
a = p₁^a₁ * p₂^a₂ * p₃^a₃ * ... * pₙ^aₙ
b = p₁^b₁ * p₂^b₂ * p₃^b₃ * ... * pₙ^bₙ

where p₁, p₂, p₃, ..., pₙ are prime numbers and a₁, a₂, a₃, ..., aₙ, b₁, b₂, b₃, ..., bₙ are their respective powers.

1. HCF Calculation:
- The HCF of a and b will be the product of the common prime factors raised to the minimum power.
- HCF(a, b) = p₁^min(a₁, b₁) * p₂^min(a₂, b₂) * p₃^min(a₃, b₃) * ... * pₙ^min(aₙ, bₙ)

2. LCM Calculation:
- The LCM of a and b will be the product of all prime factors raised to their maximum power.
- LCM(a, b) = p₁^max(a₁, b₁) * p₂^max(a₂, b₂) * p₃^max(a₃, b₃) * ... * pₙ^max(aₙ, bₙ)

Relation between HCF and LCM:
- From the above calculations, we can observe that the HCF and LCM of two numbers are related by their prime factors.
- The HCF is the product of common prime factors raised to the minimum power, while the LCM is the product of all prime factors raised to their maximum power.
- Mathematically, we can express this relationship as HCFLCM = Product of two natural numbers.

Conclusion:
The relationship between the HCF and LCM of two natural numbers is given by HCFLCM = Product of two natural numbers. This relationship is derived from the prime factorization of the numbers and helps us understand the fundamental properties of

Answer

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Answer

A is right answer

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