To find the two numbers whose Lowest Common Multiple (LCM) is 4536 and Highest Common Factor (HCF), also known as the Greatest Common Divisor (GCD), is 189, we can use the relationship between the LCM and HCF of two numbers. The product of the LCM and HCF of two numbers is equal to the product of the numbers themselves.
Mathematically, if the two numbers are \(a\) and \(b\), this relationship is given by:
\[ \text{LCM}(a, b) \times \text{HCF}(a, b) = a \times b \]
Given:
\[ \text{LCM} = 4536 \]
\[ \text{HCF} = 189 \]
We can use these values to find the product of \(a\) and \(b\):
\[ a \times b = \text{LCM} \times \text{HCF} \]
\[ a \times b = 4536 \times 189 \]
Now, let's calculate the value of this product:
\[ a \times b = 4536 \times 189 \]
\[ a \times b = 857784 \]
We now need to find two numbers that multiply to 857784 and have an LCM of 4536 and an HCF of 189. Since the HCF of the two numbers is 189, it means that the two numbers are some multiple of 189.
Let's use this information to establish the two numbers. We know that one number is a multiple of 189, and the other must also be a multiple of 189 in order for their HCF to be 189. If one of the numbers is 189 itself (the smallest multiple), then the other number has to be 857784 ÷ 189 because:
\[ \frac{a \times b}{\text{HCF}} = \frac{857784}{189} \]
Now let's calculate this:
\[ \frac{857784}{189} = 4536 \]
So the two numbers are 189 and 4536 themselves. However, we need to make sure that no numbers exist between this range that have the same HCF (189) and LCM (4536).
Since we know that any pair of numbers having 189 as their HCF will be multiples of 189, let's check the largest number that is less than 4536 and is a multiple of 189:
\[ \frac{4536}{189} \approx 24 \]
Now, we have 24 multiples to check. However, out of these multiples, we need to find the one which has 4536 as the LCM with 189. Since 4536 is a factor of 24 (189 times 24), any other factor would not have 4536 as the LCM; the LCM would instead be a smaller number.
Therefore, the only two numbers between 189 and 4536 that have an LCM of 4536 and an HCF of 189 are 189 and 4536 themselves.
