Least Common Multiple

The least common multiple (LCM) is the smallest number \( n \) that can be evenly divided by two or more given numbers, like \(a\) and \(b\), and it is always greater than zero. $$ n = \mathrm{LCM}(a, b) $$

The LCM is particularly useful for working with fractions, ratios, and divisibility problems.

To calculate the least common multiple of two numbers \(a\) and \(b\), several methods are available, but one of the most systematic and effective is prime factorization.

This method relies on the fact that any number can be broken down into a product of prime factors—numbers that are only divisible by one and themselves.

Prime Factorization Method

To find the LCM, break each number into its prime factors. The LCM is the product of all unique prime factors, each raised to its highest power across the numbers.

This method calculates the LCM by decomposing numbers into their prime factors.

For example, let’s find the LCM of 18 and 20.

First, factorize each number into its prime factors:

$$ 18 = 2 \times 3^2 $$

$$ 20 = 2^2 \times 5 $$

Now, take each unique factor \( 2, 3, 5 \), using the highest exponent from either number, and multiply them together.

$$ \mathrm{LCM}(18,20) = 2^2 \times 3^2 \times 5 = 180 $$

The least common multiple of 18 and 20 is 180, the smallest number divisible by both 18 and 20.

Why is the LCM Useful?

The least common multiple is especially helpful when working with fractions in math.

For adding or subtracting fractions with different denominators, for instance, you need a common denominator. The LCM of the denominators provides the simplest shared denominator.

Consider, for example, the sum of these fractions:

$$ \frac{5}{6} + \frac{7}{10} $$

Here, the denominators 6 and 10 don’t share any multiples directly, so we need to find their least common multiple (LCM).

Break down the denominators into their prime factors:

• 6 factors into \(2 \times 3\),
• 10 factors into \(2 \times 5\).

The LCM of 6 and 10 includes all prime factors, each taken at its highest power:

$$ \mathrm{LCM}(6, 10) = 2 \times 3 \times 5 = 30 $$

Now convert \(\frac{5}{6}\) to have a denominator of 30:

$$ \frac{5}{6} = \frac{5 \times 5}{6 \times 5} = \frac{25}{30} $$

Convert \(\frac{7}{10}\) to a denominator of 30 as well:

$$ \frac{7}{10} = \frac{7 \times 3}{10 \times 3} = \frac{21}{30} $$

With both fractions now sharing the same denominator, you can add the numerators:

$$ \frac{25}{30} + \frac{21}{30} = \frac{25 + 21}{30} = \frac{46}{30} $$

To simplify further, divide both the numerator and denominator by 2:

$$ \frac{46}{30} = \frac{23}{15} $$

So, the sum of \(\frac{5}{6}\) and \(\frac{7}{10}\) is:

$$ \frac{5}{6} + \frac{7}{10} = \frac{23}{15} $$

In summary, the LCM simplifies calculations and allows you to tackle complex problems more efficiently.

The Link Between Greatest Common Divisor and Least Common Multiple

The connection between the greatest common divisor (G.C.D.) and the least common multiple (L.C.M.) of two numbers \(a\) and \(b\) is captured in the formula: $$ a \cdot b = G.C.D.(a, b) \cdot L.C.M.(a, b) $$

In essence, this formula shows that if you know the greatest common divisor of two numbers, you can find the least common multiple with the calculation:

$$ L.C.M.(a, b) = \frac{a \cdot b}{G.C.D.(a, b)} $$

This equation applies to any pair of integers, allowing you to determine one value if you have the product and the other value.

Note that if \(a\) and \(b\) are relatively prime, their L.C.M. is simply \(a \cdot b\). In this case, the G.C.D. can only be 1.

Practical Example

Consider the numbers \(a = 15\) and \(b = 20\).

First, break each number down into its prime factors:

$$ a = 15 = 3 \cdot 5 $$

$$ b = 20 = 2^2 \cdot 5 $$

Now, find the G.C.D. by selecting the common factors with the lowest exponents:

$$ G.C.D.(15, 20) = 5 $$

Next, calculate the L.C.M. by including all factors with the highest exponents:

$$ L.C.M.(15, 20) = 2^2 \cdot 3 \cdot 5 = 60 $$

Now, let's check the relationship between the G.C.D. and L.C.M.:

$$ a \cdot b = G.C.D.(a, b) \cdot L.C.M.(a, b) $$

$$ 15 \cdot 20 = 5 \cdot 60 $$

Both sides are equal to 300, confirming that \( a \cdot b = G.C.D.(a, b) \cdot L.C.M.(a, b) \).

So, if you know the greatest common divisor of two numbers, like \( G.C.D.(15, 20) = 5 \), you can also find the least common multiple:

$$ L.C.M.(15, 20) = \frac{15 \cdot 20}{G.C.D.(15, 20)} = \frac{15 \cdot 20}{5} = 3 \cdot 20 = 60 $$

This method provides an alternative way to calculate the L.C.M. by using the G.C.D.