To find the least common denominator (LCD) of the fractions [tex]\(\frac{5}{6}\)[/tex] and [tex]\(\frac{4}{7}\)[/tex], let's follow these steps:
1. Identify the denominators of both fractions. The denominators are 6 and 7.
2. Find the Least Common Multiple (LCM) of the two denominators. The LCM of two numbers is the smallest number that is a multiple of both of the numbers. In other words, it's the least common multiple that both denominators can divide into without leaving a remainder.
3. To find the LCM of the two denominators, we can use the formula:
[tex]\[
\text{LCM}(a, b) = \frac{|a \times b|}{\text{GCD}(a, b)}
\][/tex]
where [tex]\(a = 6\)[/tex] and [tex]\(b = 7\)[/tex], and GCD stands for Greatest Common Divisor.
4. Compute the Greatest Common Divisor (GCD) of the two numbers. Since 6 and 7 are both prime relative to each other (they share no common divisors other than 1),
[tex]\[
\text{GCD}(6, 7) = 1
\][/tex]
5. Substitute the values into the LCM formula:
[tex]\[
\text{LCM}(6, 7) = \frac{|6 \times 7|}{1} = 42
\][/tex]
6. The LCM of 6 and 7 is 42.
Therefore, the least common denominator for the fractions [tex]\(\frac{5}{6}\)[/tex] and [tex]\(\frac{4}{7}\)[/tex] is 42.
Answer: 42
