Finding the LCM of Two Numbers using HCF and Ratio

This problem requires us to find the Least Common Multiple (LCM) of two numbers when their Highest Common Factor (HCF) and the ratio between the numbers are given. We can use the fundamental relationship between two numbers, their HCF, and their LCM to solve this.

Understanding the Relationship between Numbers, HCF, and LCM

For any two positive integers, the product of the numbers is equal to the product of their HCF and LCM.

Mathematically, this can be written as:

$\text{Number}_1 \times \text{Number}_2 = \text{HCF} \times \text{LCM}$

Given Information

• HCF of the two numbers = 9
• Ratio of the two numbers = 14 : 19

Representing the Numbers

When the ratio of two numbers is given as $a : b$ and their HCF is $h$, the numbers can be represented as $a \times h$ and $b \times h$. This is because the HCF is the greatest common divisor that has been divided out to get the ratio. Thus, multiplying the ratio terms by the HCF gives back the original numbers.

In this case, the ratio is 14 : 19 and the HCF is 9.

• First Number = Ratio term 1 $\times$ HCF = $14 \times 9$
• Second Number = Ratio term 2 $\times$ HCF = $19 \times 9$

Calculating the Numbers

• First Number = $14 \times 9 = 126$
• Second Number = $19 \times 9 = 171$

So, the two numbers are 126 and 171. We can verify their HCF: The factors of 126 are 1, 2, 3, 6, 7, 9, 14, 18, 21, 42, 63, 126. The factors of 171 are 1, 3, 9, 19, 57, 171. The greatest common factor is 9, which matches the given HCF.

Calculating the LCM

Now we use the relationship:

$\text{Number}_1 \times \text{Number}_2 = \text{HCF} \times \text{LCM}$

Substitute the values we know:

$126 \times 171 = 9 \times \text{LCM}$

To find the LCM, divide the product of the numbers by the HCF:

$\text{LCM} = \frac{126 \times 171}{9}$

We can simplify the calculation by dividing 126 by 9:

$\frac{126}{9} = 14$

So, the equation becomes:

$\text{LCM} = 14 \times 171$

Now, calculate the product:

$14 \times 171 = 14 \times (100 + 70 + 1) = 1400 + 980 + 14 = 2380 + 14 = 2394$

Alternatively, there is a direct formula for LCM when the numbers are given in ratio $a:b$ and HCF $h$: $\text{LCM} = h \times a \times b$.

Using this formula:

$\text{LCM} = 9 \times 14 \times 19$

First, multiply 14 by 19:

$14 \times 19 = 14 \times (20 - 1) = 280 - 14 = 266$

Now, multiply the result by 9:

$\text{LCM} = 9 \times 266 = 9 \times (200 + 60 + 6) = 1800 + 540 + 54 = 2340 + 54 = 2394$

Both methods yield the same result.

Conclusion

The LCM of the two numbers is 2394.

Revision Table: HCF and LCM Concepts

Additional Information: HCF, LCM, and Ratios

When two numbers are in the ratio $a:b$ (where $a$ and $b$ are coprime, meaning their HCF is 1), and their HCF is $h$, the actual numbers are $ah$ and $bh$.

Using the fundamental relationship:

$(ah) \times (bh) = h \times \text{LCM}$

$ab \times h^2 = h \times \text{LCM}$

Dividing both sides by $h$ (assuming $h \ne 0$):

$\text{LCM} = ab \times h$

This derived formula, $\text{LCM} = \text{HCF} \times \text{Ratio term}_1 \times \text{Ratio term}_2$, provides a shortcut to calculate the LCM directly from the HCF and the ratio, assuming the ratio terms are coprime. In our problem, the ratio is 14:19. 14 and 19 are coprime (factors of 14 are 1, 2, 7, 14; factors of 19 are 1, 19; common factor is only 1). So, this formula is applicable.

Using the direct formula $\text{LCM} = h \times a \times b$ with $h=9$, $a=14$, $b=19$:

$\text{LCM} = 9 \times 14 \times 19 = 9 \times 266 = 2394$.

This confirms the result obtained by first finding the numbers.