Calculates the greatest common divisor (GCD) and least common multiple (LCM) of all the values present in its arguments.
Details
The computation is based on the Euclidean algorithm without using the
extended version. The greatest common divisor for all numbers in the integer
vector x will be computed (the multiple GCD). Negative values are
allowed and enter via their absolute value; logical vectors are coerced
to integer.
Note
The following relation is always true:
n * m = GCD(n, m) * LCM(n, m)
It also holds when one of the values is zero, and that is the shortest way
to see why LCM(0, 6) has to be 0 rather than 6.
Zero
Zero behaves differently in the two functions, which is why they do not
treat it the same way. For the greatest common divisor it is
neutral - every number divides 0, so GCD(0, a) is
abs(a) and zeros can simply be dropped. For the least common
multiple it is absorbing - 0 is a multiple of every number and the
smallest non-negative one, so LCM(0, a) is 0. GCD(0, 0) and
LCM(0, 0) are both 0.
See also
Other number.theory:
digitSum(),
divisors(),
factorize(),
fibonacci(),
isOdd(),
isPrime(),
primes()
Examples
GCD(12, 10)
#> [1] 2
GCD(144, 233) # Fibonacci numbers are relatively prime to each other
#> [1] 1
LCM(12, 10)
#> [1] 60
LCM(144, 233) # = 144 * 233
#> [1] 33552
# all elements will be flattened by unlist
GCD(2, 3, c(5, 7) * 11)
#> [1] 1
GCD(c(2*3, 3*5, 5*7))
#> [1] 1
LCM(c(2, 3, 5, 7) * 11)
#> [1] 2310
LCM(2*3, 3*5, 5*7)
#> [1] 210
# zero is neutral for the GCD and absorbing for the LCM
GCD(0, 6)
#> [1] 6
LCM(0, 6)
#> [1] 0
# n * m == GCD(n, m) * LCM(n, m), zero included
GCD(0, 6) * LCM(0, 6)
#> [1] 0