Two numbers are in the ratio 6:3. If their HCF is 3, what is the LCM?
Understand the Problem
The question states that two numbers are in the ratio 6:3. It provides that their highest common factor (HCF) is 3. The task is to find the least common multiple (LCM) of these two numbers. The first step will be finding the two numbers, followed by calculating the LCM.
Answer
The LCM is 6.
Answer for screen readers
The least common multiple (LCM) of the two numbers is 6.
Steps to Solve
- Express the numbers in terms of the ratio and HCF
Since the numbers are in the ratio 6:3, we can represent them as $6x$ and $3x$. Given that their HCF is 3, we can deduce that $x = 1$ because $HCF(6, 3) = 3$. Thus, the numbers are $6 \times 1 = 6$ and $3 \times 1 = 3$.
- Calculate the LCM of the two numbers
To find the LCM of 6 and 3, we can list the multiples of each number. Multiples of 6: 6, 12, 18, 24, ... Multiples of 3: 3, 6, 9, 12, ... The smallest common multiple is 6.
Alternatively, we can use the formula $LCM(a, b) = \frac{|a \times b|}{HCF(a, b)}$. $LCM(6, 3) = \frac{|6 \times 3|}{3} = \frac{18}{3} = 6$.
The least common multiple (LCM) of the two numbers is 6.
More Information
The Least Common Multiple (LCM) is the smallest positive integer that is divisible by both numbers.
Tips
A common mistake is to assume $x$ is always 1 when the HCF is given. Always check if the ratio is already in its simplest form. In this case, the ratio 6:3 simplifies to 2:1. If the ratio was given as 12:6 and the HCF was 6, then the numbers would be $12 \times 1 = 12$ and $6 \times 1 = 6$, and the simplified ratio would be 2:1. Another common mistake is to incorrectly calculate the LCM, especially when dealing with larger numbers. Using the formula involving HCF can help avoid this.
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