GCF and LCM: Practice

GCF is short for greatest common factor: the largest number that is a factor of two given numbers.

$\operatorname{GCF}(a, b) = \text{largest number dividing both } a \text{ and } b$

It is always a factor of both numbers, so it is never larger than either one.

List the factors of each, then keep the ones in both lists. Factors of $8$: $1, 2, 4, 8$. Factors of $12$: $1, 2, 3, 4, 6, 12$.

$\text{in both lists: } 1, 2, 4$

So the common factors of $8$ and $12$ are $1, 2,$ and $4$.

Factors of $10$: $1, 2, 5, 10$. Factors of $15$: $1, 3, 5, 15$. The common factors are $1$ and $5$.

$\text{greatest common factor} = 5$

So $5$ is the largest number dividing both $10$ and $15$.

The positive multiples of $5$ are what you get by multiplying $5$ by $1, 2, 3, 4, \ldots$

$5 \times 1 = 5, \quad 5 \times 2 = 10, \quad 5 \times 3 = 15, \quad 5 \times 4 = 20$

So the first four positive multiples are $5, 10, 15, 20$. Counting would include $0$, but $0$ is not positive, so it is excluded here.

Multiples of $2$: $2, 4, 6, 8, \ldots$ Multiples of $3$: $3, 6, 9, \ldots$ The first value in both lists is $6$.

$\operatorname{LCM}(2, 3) = 6$

This is the soonest counting by twos and by threes lands on the same number.

Multiples of $3$: $3, 6, 9, 12, \ldots$ Multiples of $4$: $4, 8, 12, \ldots$ The first shared value is $12$.

$\operatorname{LCM}(3, 4) = 12$

Since $3$ and $4$ share no factor, the LCM equals their product.

Multiples of $4$: $4, 8, 12, \ldots$ Multiples of $8$: $8, 16, \ldots$ The first value in both is $8$.

$\operatorname{LCM}(4, 8) = 8$

When the smaller number divides the larger one, the larger number is the LCM.

Factors of $9$: $1, 3, 9$. Factors of $12$: $1, 2, 3, 4, 6, 12$. The common factors are $1$ and $3$.

$\text{greatest common factor} = 3$

The value $36$ is the LCM of $9$ and $12$, not the GCF.

Every whole number is divisible by $1$, so $1$ is a factor of every number.

$1 \text{ divides } a \quad \text{and} \quad 1 \text{ divides } b$

That makes $1$ a common factor of any pair, which is why the GCF is at least $1$ and the list of common factors is never empty.

Factors of $5$: $1, 5$. Factors of $11$: $1, 11$. The only factor they share is $1$.

$\text{greatest common factor} = 1$

Two different primes share no factor besides $1$, so their GCF is $1$.

Multiples of $5$: $5, 10, 15, \ldots$ Multiples of $10$: $10, 20, \ldots$ The first shared value is $10$.

$\operatorname{LCM}(5, 10) = 10$

Since $5$ divides $10$, the larger number $10$ is the LCM.

The factors of $12$ are $1, 2, 3, 4, 6, 12$, and every one of them is shared with itself. The largest factor of $12$ is $12$.

$\operatorname{GCF}(12, 12) = 12$

The GCF of a number with itself is the number, since it divides itself.