9. The Highest Common Factor (HCF) of two numbers is 6 . The Lowest Common Multiple (LCM) of the same numbers is 60 . What are the two numbers?

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To find the two numbers given their Highest Common Factor (HCF) and Lowest Common Multiple (LCM), we can use the relationship between HCF, LCM, and the two numbers $$ a $$ and $$ b $$:

$$
\text{HCF}(a, b) \times \text{LCM}(a, b) = a \times b
$$

Given:
- HCF = 6
- LCM = 60

We can set up the equation:

$$
6 \times 60 = a \times b
$$

Now, let's calculate $$ a \times b $$:
Calculate the value by following steps:
- step0: Calculate:
  $$6\times 60$$
- step1: Multiply the numbers:
  $$360$$
Now we have:

$$
a \times b = 360
$$

Next, we can express the two numbers $$ a $$ and $$ b $$ in terms of their HCF. Since the HCF is 6, we can write:

$$
a = 6m \quad \text{and} \quad b = 6n
$$

where $$ m $$ and $$ n $$ are coprime integers (i.e., their HCF is 1). Substituting these into the product equation gives:

$$
(6m) \times (6n) = 360
$$

This simplifies to:

$$
36mn = 360
$$

Dividing both sides by 36:

$$
mn = 10
$$

Now we need to find pairs of coprime integers $$ (m, n) $$ such that their product is 10. The pairs are:

1. $$ (1, 10) $$
2. $$ (2, 5) $$

Now we can find the corresponding values of $$ a $$ and $$ b $$ for each pair:

1. For $$ (m, n) = (1, 10) $$:
   - $$ a = 6 \times 1 = 6 $$
   - $$ b = 6 \times 10 = 60 $$

2. For $$ (m, n) = (2, 5) $$:
   - $$ a = 6 \times 2 = 12 $$
   - $$ b = 6 \times 5 = 30 $$

Thus, the two pairs of numbers that satisfy the conditions are $$ (6, 60) $$ and $$ (12, 30) $$.

To summarize, the two numbers are:
- $$ 6 $$ and $$ 60 $$
- $$ 12 $$ and $$ 30 $$
The two numbers are 6 and 60, or 12 and 30.

The Highest Common Factor (HCF) of two numbers is 6 .
The Lowest Common Multiple (LCM) of the same numbers is 60 .
What are the two numbers?

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