To solve this problem, let's follow a step-by-step approach to find the value of [tex]\( k \)[/tex]:
1. We are given the Highest Common Factor (HCF) of the two numbers 2520 and 6600, which is 120.
2. We also know that the product of the HCF and the Least Common Multiple (LCM) of two numbers is equal to the product of the two numbers themselves. In other words, the relationship can be expressed as:
[tex]\[
\text{HCF} \times \text{LCM} = \text{num1} \times \text{num2}
\][/tex]
Here, [tex]\(\text{num1} = 2520\)[/tex] and [tex]\(\text{num2} = 6600\)[/tex].
3. Plugging in the given values, we get:
[tex]\[
120 \times \text{LCM} = 2520 \times 6600
\][/tex]
4. To find the LCM, we solve for it as:
[tex]\[
\text{LCM} = \frac{2520 \times 6600}{120}
\][/tex]
5. Simplifying the right-hand side, we find:
[tex]\[
\text{LCM} = \frac{16632000}{120} = 138600
\][/tex]
6. We are also given that the LCM can be expressed in the form [tex]\(252 \times k\)[/tex]. So, we equate this to our calculated LCM:
[tex]\[
252 \times k = 138600
\][/tex]
7. To find [tex]\( k \)[/tex], we solve for it by dividing both sides by 252:
[tex]\[
k = \frac{138600}{252}
\][/tex]
8. Simplifying this division, we get:
[tex]\[
k = 550
\][/tex]
Therefore, the value of [tex]\( k \)[/tex] is [tex]\( 550 \)[/tex].
