### Numbers: Integers

### Greatest common divisor and least common multiple

When we have two integers, it can be useful to look at the common divisors of these two integers.

The common divisors of two integers are all integers that are divisors of both integers.

The **greatest common divisor** of two integers is the largest integer of all shared divisors of these two integers.

We abbreviate the greatest common divisor as #\mathrm{gcd}#.

**Example**

Common divisors of #40# and #160#:

#1#, #2#, #4#, #5#, #8#, #10#, #20#, #40#

So #\mathrm{gcd}(40,160)=40#

Besides considering the divisors, it can also be useful to look at common multiples of two integers.

Least common multiple

The multiples of #3# are #3,6,9,12,15,\dots#

So, a multiple of #3# is an integer that can be divided by #3#.

A common multiple of two integers is an integer that is divisible by both integers. An easy-to-find common multiple of two integers, is their product.

The **least common multiple** of two integers is the smallest, positive integer within the common multiples of these two integers.

We abbreviate the least common multiple as #\mathrm{lcm}#.

**Example**

Multiples of #4#:

#4#, #8#, #12#, #16#, #\ldots#

Multiples of #6#:

#6#, #12#, #18#, #24#, #\ldots#

Common multiples:

#12#, #24#, #36#, #48#, #\ldots#

So #\mathrm{lcm}(4,6)=12#

The divisors of #27# are: # 1 , 3 , 9 , 27 #.

The divisors of #18# are: # 1 , 2 , 3 , 6 , 9 , 18 #.

Thus, the common divisors are: # 1 , 3 , 9 #.

The greatest common divisor is #9#.

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