We are given the following information:

• The Highest Common Factor (HCF) of two numbers is 175.
• The Lowest Common Multiple (LCM) of these two numbers is 12,600.
• Both numbers are greater than the HCF.

We can use the relationship between the HCF, LCM, and the product of two numbers to solve this problem. The formula that relates these quantities is:

$\text{HCF} \times \text{LCM} = \text{Product of the two numbers}$

Let's denote the two numbers as $a$ and $b$. From the formula, we have:

$175 \times 12,600 = a \times b$

Calculating the product:

$175 \times 12,600 = 2,205,000$

So,

$a \times b = 2,205,000$

Next, since both $a$ and $b$ are greater than 175, they must be multiples of 175. Let’s express $a$ and $b$ as:

$a = 175x \quad \text{and} \quad b = 175y$

where $x$ and $y$ are integers greater than 1 and $\text{HCF}(x, y) = 1$ (since the HCF of $a$ and $b$ is 175). Substituting these into the product equation:

$(175x) \times (175y) = 2,205,000$

This simplifies to:

$175^2 \times xy = 2,205,000$

$30,625 \times xy = 2,205,000$

Now, divide both sides by 30,625:

$xy = \frac{2,205,000}{30,625} = 72$

So, $x \times y = 72$. Now we need to find two numbers $x$ and $y$ such that:

• $x \times y = 72$
• $\text{HCF}(x, y) = 1$

The pairs of integers that multiply to give 72 and have an HCF of 1 are:

• $(8, 9)$

Thus, $x = 8$ and $y = 9$. Therefore, the two numbers $a$ and $b$ are:

$a = 175 \times 8 = 1,400$ $b = 175 \times 9 = 1,575$

Thus, the two numbers are 1,400 and 1,575.

Would you like more details, or do you have any questions?

Here are 5 related questions:

1. How do we find the HCF and LCM of two numbers manually?
2. What is the significance of prime factorization in finding the HCF and LCM?
3. Can two numbers have an HCF greater than 1 and still be co-prime?
4. How does the product of two numbers relate to their HCF and LCM?
5. What is the difference between the greatest common divisor and least common multiple?

Tip: If two numbers have a known HCF and LCM, their product will always equal the product of the HCF and LCM.