Lesson 30: Least Common Multiple (LCM)

New Concept

Reciprocals are two numbers whose product is 1. To find a fraction's reciprocal, you simply flip its numerator and denominator.

Reciprocals are two numbers whose product is 1.

$$5 \times \frac{1}{5} = 1$$

To form the reciprocal of a fraction, we reverse the terms of the fraction.

$$\frac{7}{8} \quad \longleftrightarrow \quad \frac{8}{7}$$

What’s next

Now, let's put this into practice. You’ll tackle worked examples on finding reciprocals for different numbers and using them to solve equations.

Property

The smallest positive number that is a multiple of two or more numbers is the least common multiple (LCM).

Examples

• The LCM of 3 and 4 is 12, since their multiples are (3, 6, 9, 12) and (4, 8, 12).
• The LCM of 2 and 4 is 4, since their multiples are (2, 4, 6) and (4, 8, 12).
• The LCM of 4 and 6 is 12, since their multiples are (4, 8, 12) and (6, 12, 18).

Explanation

Imagine two friends racing on a track, but running at different speeds. The LCM is like the first spot on the track where they meet up again after starting! It's the smallest number that both of their numbers can 'land' on perfectly when you count up.

Property

Reciprocals are two numbers whose product equals 1. To find the reciprocal of a fraction, you reverse its terms. The reciprocal of $\frac{a}{b}$ is $\frac{b}{a}$.

Examples

• The reciprocal of 5 is $\frac{1}{5}$, because you can write 5 as $\frac{5}{1}$ and then flip it.
• The reciprocal of $\frac{3}{4}$ is $\frac{4}{3}$, because $\frac{3}{4} \times \frac{4}{3} = \frac{12}{12} = 1$.
• To solve the equation $\frac{5}{6} \times \square = 1$, you use the reciprocal of $\frac{5}{6}$, which is $\frac{6}{5}$.

Explanation

Think of a reciprocal as a number's 'upside-down' twin! When you multiply a number by its reciprocal, they team up to become 1. It's a handy trick for division problems. For any fraction, just flip the numerator and the denominator to find its twin.