Finding the LCM of 6.3 and 0.07

The question asks us to find the Least Common Multiple (LCM) of the two decimal numbers, $6.3$ and $0.07$. We are given four options: $63$, $0.63$, $6.3$, and $0.063$.

Understanding Least Common Multiple (LCM)

The Least Common Multiple (LCM) of two or more numbers is the smallest positive number that is a multiple of all the numbers involved. When dealing with decimal numbers, it's often helpful to convert them into a more manageable format, like fractions or whole numbers, to calculate the LCM.

Method 1: Converting Decimals to Fractions

One effective way to find the LCM of decimal numbers is to convert them into fractions first.

• First, express $6.3$ as a fraction: $6.3 = \frac{63}{10}$.
• Next, express $0.07$ as a fraction: $0.07 = \frac{7}{100}$.

Now, we need to find the LCM of these two fractions: $\frac{63}{10}$ and $\frac{7}{100}$. The formula for the LCM of two fractions $\frac{a}{b}$ and $\frac{c}{d}$ is:

$ LCM\left(\frac{a}{b}, \frac{c}{d}\right) = \frac{LCM(a, c)}{GCD(b, d)} $

Here, $a = 63$, $b = 10$, $c = 7$, and $d = 100$. We need to calculate the LCM of the numerators ($63$ and $7$) and the Greatest Common Divisor (GCD) of the denominators ($10$ and $100$).

• Calculating LCM(63, 7): We list multiples of the larger number (63): 63, 126, ... We list multiples of the smaller number (7): 7, 14, ..., 56, 63, ... The smallest number that appears in both lists is $63$. So, $LCM(63, 7) = 63$.
• Calculating GCD(10, 100): We find the factors of 10 (1, 2, 5, 10) and the factors of 100 (1, 2, 4, 5, 10, 20, 25, 50, 100). The largest factor common to both is $10$. So, $GCD(10, 100) = 10$.

Now, applying the formula for the LCM of fractions:

$ LCM\left(\frac{63}{10}, \frac{7}{100}\right) = \frac{LCM(63, 7)}{GCD(10, 100)} = \frac{63}{10} $

Finally, we convert the resulting fraction $\frac{63}{10}$ back into a decimal:

$ \frac{63}{10} = 6.3 $

Method 2: Converting Decimals to Integers

An alternative method involves converting the decimal numbers into integers, finding their LCM, and then adjusting the result.

• First, determine the maximum number of digits after the decimal point in the given numbers. $6.3$ has one digit after the decimal point, and $0.07$ has two. The maximum is 2.
• Multiply both original numbers by $10$ raised to the power of the maximum number of decimal places (which is 2), so we multiply by $10^2 = 100$.
  • $6.3 \times 100 = 630$
  • $0.07 \times 100 = 7$
  These are now whole numbers.
• Calculate the LCM of these integers, $630$ and $7$.
  • We can see that $630$ is divisible by $7$ ($630 \div 7 = 90$). When one number is a multiple of another, their LCM is the larger number.
  • Therefore, $LCM(630, 7) = 630$.
• To find the LCM of the original decimal numbers, divide the LCM of the integers ($630$) by the multiplier we used ($100$):

  $ \frac{630}{100} = 6.3 $

Final Answer Calculation

Both methods confirm that the Least Common Multiple of $6.3$ and $0.07$ is $6.3$.