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 #35# are: # 1 , 5 , 7 , 35 #.
The divisors of #12# are: # 1 , 2 , 3 , 4 , 6 , 12 #.
Thus, the common divisors are: #1#.
The greatest common divisor is #1#.
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