If $a=6$ and $b=17$, then evaluate HCF $(a,b) imes LCM$ (a, b):

Question

If $a=6$ and $b=17$, then evaluate HCF $(a,b)	imes LCM$ (a, b):

Accepted Answer

The correct option is 2:

To solve this problem, we need to recall a fundamental property relating the Highest Common Factor (HCF) and Least Common Multiple (LCM) of two numbers.

Key Concept: For any two positive integers, say $a$ and $b$, the product of their HCF and LCM is always equal to the product of the numbers themselves.
That is, $	ext{HCF}(a, b) 	imes 	ext{LCM}(a, b) = a 	imes b$.

Given:
The first number, $a = 6$.
The second number, $b = 17$.

We need to evaluate $	ext{HCF}(a, b) 	imes 	ext{LCM}(a, b)$.

Using the property mentioned above:
$	ext{HCF}(6, 17) 	imes 	ext{LCM}(6, 17) = 6 	imes 17$

Now, let's calculate the product of $a$ and $b$:
$6 	imes 17 = 102$

Therefore, $	ext{HCF}(6, 17) 	imes 	ext{LCM}(6, 17) = 102$.

Let's also quickly verify the HCF and LCM for $a=6$ and $b=17$:
Prime factorization of $6 = 2 	imes 3$.
Prime factorization of $17 = 17$ (since 17 is a prime number).

Since 6 and 17 have no common prime factors other than 1, their Highest Common Factor (HCF) is 1.
$	ext{HCF}(6, 17) = 1$.

The Least Common Multiple (LCM) of two prime numbers or two co-prime numbers is their product. Since 6 and 17 are co-prime (their HCF is 1), their LCM will be their product.
$	ext{LCM}(6, 17) = 6 	imes 17 = 102$.

Now, multiplying HCF and LCM:
$	ext{HCF}(6, 17) 	imes 	ext{LCM}(6, 17) = 1 	imes 102 = 102$.

Both methods confirm that the value is 102.

The incorrect option is 1: This option gives 60. This could be a result of an incorrect calculation or a misunderstanding of the relationship between HCF, LCM, and the numbers themselves.
The incorrect option is 3: This option gives 617. This might be a result of concatenating the numbers or some other unrelated calculation.
The incorrect option is 4: This option gives 96. This is an incorrect calculation.