We are given the following details:

• The highest common factor (HCF) of two numbers is 14.
• The lowest common multiple (LCM) of the same two numbers is 210.
• Both numbers are less than 100.
• We need to find the difference between the two numbers.

Relationship Between HCF and LCM:

For two numbers $a$ and $b$, the product of their HCF and LCM is equal to the product of the numbers: $HCF(a, b) \times LCM(a, b) = a \times b$ In this case: $14 \times 210 = a \times b$ $a \times b = 2940$

Thus, the product of the two numbers is 2940. Now, we need to find two numbers that satisfy:

• The product is 2940.
• The HCF is 14.
• Both numbers are less than 100.

Express the numbers in terms of the HCF:

Let the two numbers be $14x$ and $14y$, where $x$ and $y$ are coprime (since the HCF is 14). Then: $14x \times 14y = 2940$ $196xy = 2940$ $xy = \frac{2940}{196} = 15$

So, $x \times y = 15$. The coprime pairs of numbers whose product is 15 are: $(1, 15), (3, 5)$

Calculate the Numbers:

• For $(x, y) = (1, 15)$, the numbers are $14 \times 1 = 14$ and $14 \times 15 = 210$ (but 210 is greater than 100, so discard this).
• For $(x, y) = (3, 5)$, the numbers are $14 \times 3 = 42$ and $14 \times 5 = 70$.

Difference Between the Numbers:

The two numbers are 42 and 70. The difference is: $70 - 42 = 28$

Thus, the difference between the two numbers is 28.

Would you like more details or have any questions?

Here are 5 related questions to explore:

1. How do you calculate the least common multiple (LCM) of two numbers?
2. What is the relationship between the HCF and LCM of two numbers?
3. How can we find coprime numbers in a given set?
4. What are other methods to find the difference between two numbers given their HCF and LCM?
5. How would the solution change if the numbers were not restricted to being less than 100?

Tip: When dealing with HCF and LCM, it's helpful to remember that the product of HCF and LCM is always equal to the product of the two numbers.