Three numbers are in the ratio of 3:4:5 and their LCM is 2400. What is their HCF?
Understand the Problem
The question is asking for the highest common factor (HCF) of three numbers that are in the ratio of 3:4:5, given that their least common multiple (LCM) is 2400. We need to use the relationship between LCM, HCF, and the product of the numbers to find the HCF.
Answer
$1600$
Answer for screen readers
The highest common factor (HCF) is $1600$.
Steps to Solve
- Define the Numbers Based on the Ratio
Let the three numbers be $3x$, $4x$, and $5x$ where $x$ is a common factor.
- Use the Relationship Between LCM and HCF
The relationship states that:
$$ \text{LCM}(a, b, c) \cdot \text{HCF}(a, b, c) = a \cdot b \cdot c $$
Here, we know:
- LCM = 2400
- The three numbers are $3x$, $4x$, and $5x$.
- Calculate the Product of the Numbers
The product of the three numbers can be calculated as follows:
$$ (3x) \cdot (4x) \cdot (5x) = 60x^3 $$
- Set up the Equation
Using the relationship, set up the equation:
$$ 2400 \cdot \text{HCF} = 60x^3 $$
- Solve for HCF
Let HCF be denoted as $h$:
$$ h = \frac{60x^3}{2400} $$ $$ h = \frac{x^3}{40} $$
- Find x Using the LCM
Next, we need to find $x$. To find $x$, we start by determining the LCM of the three numbers:
The LCM of $3x$, $4x$, and $5x$ is given by:
$$ \text{LCM}(3, 4, 5) \cdot x = 60x $$
Now, we can set the LCM equal to 2400:
$$ 60x = 2400 $$
- Calculate x
Solve for $x$:
$$ x = \frac{2400}{60} = 40 $$
- Substitute x Back to Find HCF
Using the value of $x$, find HCF:
$$ h = \frac{x^3}{40} = \frac{40^3}{40} = 40^2 = 1600 $$
Now we have the highest common factor.
- Final Result
Thus, the highest common factor is $h = 1600$.
The highest common factor (HCF) is $1600$.
More Information
The HCF is a vital concept in number theory as it represents the greatest number that divides two or more numbers without leaving a remainder. In problems involving ratios and their multiples, using the relationships between LCM and HCF can simplify calculations significantly.
Tips
- Confusing between HCF and LCM may lead to wrong calculations.
- Not setting up the correct product of the numbers based on their ratios can cause errors.