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Least Common Multiples

Discrete Mathematics Greatest Common Factors, Least Common Multiples, and Coprimality Least Common Multiples

Least Common Multiple

Let $a$ and $b$ be integers. The least common multiple (LCM) of $a$ and $b$, denoted $lcm(a,b)$, is the smallest integer $n$ such that both $a|n$ and $b|n$.

Least common multiples are somewhat the inverse idea of “greatest common divisors.” With greatest common divisors, we are looking for the largest number that divides two given numbers. With least common multiples, however, we are looking for a single number that our two given number divides. The following drawing helps illustrate the difference just described.

By one of our divisibility theorems, the above diagram implies that GCD divides LCM.

It is worth stating that the product of $a$ and $b$ is certainly a common multiple of $a$ but not necessarily the least common multiple. One must try to find the absolute smallest number that both $a$ and $b$ both divide. Sometimes it is a good idea to make a list of multiples of $a$ and multiples of $b$, and find the smallest number in both lists.

Another way of finding LCMs is by considering the prime factors of both numbers $a$ and $b$, finding the smallest number that has all the same prime factors of both $a$ and $b$. More on this in the next topic.

$lcm(5,10)=10$

Note that both $5$ and $10$ divide $10$, and that the multiples of $5$ are, in increasing order, $5,10,15, 20$. $10$ doesn’t divide $5$, so it follows that $10$ must be the LCM of $5$ and $10$.

Shorter explanation: If you list the multiples of $5$ and $10$, the number $10$ is the smallest number in both lists.

$lcm(20,30)=60$

Note that the multiples of $20$ are, in order: $20,40,60, 80,….$

Note that the multiples of $30$ are, in order: $30,60, 90,…$.

The smallest multiple in both lists is $60$. This is the LCM of $20$ and $30$ because it is therefore the smallest number that both $20$ and $30$ divide.

$lcm(12,40)=120$

Note that the multiples of $12$ are, in order: $12,24, 36, 48, 60, …$

Note that the multiples of $40$ are, in order: $40,80,120,…$.

The smallest multiple in both lists is $120$. This is the LCM of $12$ and $40$ because it is therefore the smallest number that both $12$ and $40$ divide.

$lcm(11,12)=132$

Note that the multiples of $11$ are, in order: $11,22,33,44,55,66,77,88,99,110,121,132,….$

Note that the multiples of $12$ are, in order: $12, 24,36,48,60,72,84,96,108, 120, 132, …$.

The smallest multiple in both lists is $132$. This is the LCM of $11$ and $12$ because it is therefore the smallest number that both $11$ and $12$ divide.

NOTICE: $11$ and $12$ are coprime and $lcm(11,12)=11\cdot 12=132$. This is of interest and will be discussed in more detail in the next topic.

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