Optimizing finding and counting proper divisors in ruby - ruby-on-rails

Question

This code needs to run under 7000ms or it times out and I am trying to learn ruby so I am here to see if anyone has any ideas that could optimize this code. Or if you can just let me know which functions in this code take the most time so I can concentrate on the parts that will do the most good.
The questions to solve is that you have to tell if the number of divisors for any umber is odd or even.
For n=12 the divisors are [1,2,3,4,6,12] – 'even'
For n=4 the divisors are [1,2,4] – 'odd'
Any help is greatly appreciated,
Thanks.
def oddity(n)
div(n) % 2 == 0 ? (return 'even'): (return 'odd')
end
def div(num)
divs = []
(1..num).each{|x| if (num % x == 0) then divs << x end}
return divs.length
end

Answer 1

The key observation here is that you need only the number of divisors, rather than the divisors themselves. Thus, a fairly simple solution is to decompose the number to primes, and check how many combinations can we form.
require 'mathn'
def div(num)
num.prime_division.inject(1){ |prod, n| prod *= n[1] + 1 }
end
prime_division returns a list of pairs, where the first is the prime and the second is its exponent. E.g.:
12.prime_division
=> [[2, 2], [3, 1]]
We simply multiply the exponents, adding 1 to each, to account for the case where this prime wasn't taken.

Answer 2

Since performance is an issue, let's compare the OP's solution with #standelaune's and #dimid's.
require 'prime'
require 'fruity'
n = 100_000
m = 20
tst = m.times.map { rand(n) }
#=> [30505, 26103, 53968, 24108, 78302, 99141, 22816, 67504, 10149, 28406,
# 18294, 92203, 73157, 5444, 24928, 65154, 24850, 64219, 68310, 64951]
def op(num) # Alex
divs = []
(1..num).each { |x| if (num % x == 0) then divs << x end }
divs.length
end
def test_op(tst) # Alex
tst.each { |n| op(n) }
end
def pd(num) # divid
num.prime_division.inject(1){ |prod, n| prod *= n[1] + 1 }
end
def test_pd(tst) #divid
tst.each { |n| nfacs_even?(n) }
end
def div(num) # standelaune
oddity = false
(1..num).each{|x| if (num % x == 0) then oddity = !oddity end}
oddity ? "odd" : "even"
end
def test_div(tst) # standelaune
tst.each { |n| div(n) }
end
compare do
_test_op { test_op tst }
_test_div { test_div tst }
_test_pd { test_pd tst }
end
Running each test 16 times. Test will take about 56 seconds.
_test_pd is faster than _test_div by 480x ± 100.0
_test_div is similar to _test_op
I'm not suprised that divid's method smokes the others, as prime_division uses (an instance of) the default prime generator, Prime::Generator23, That generator is coded in C and is fast relative to other generators in Prime subclasses.

Answer 3

You could solve this by optimising your algorithm.
You don't have to check all numbers below the number you are examining. It is enough to split your number in to it´s prime components. Then it is a simple matter of combinatorics to determine how many possible divisors there are.
One way to get all prime components could be:
PRIME_SET = [2,3,5,7,11,13,17,19]
def factorize(n)
cut_off = Math.sqrt(n)
parts = []
PRIME_SET.each do |p|
return parts if p > cut_off
if n % p == 0
n = n/p
parts << p
redo
end
end
raise 'To large number for current PRIME_SET'
end
Then computing the number of possible can be done in a number of different ways and there are probably ways of doing it without even computing them. But here is a naive implementation.
def count_possible_divisors(factors)
divisors = Set.new
(1..factors.length-1).each do |i|
factors.combination(i).each do |comb|
divisors.add(comb.reduce(1, :*))
end
end
divisors.length + 2 # plus 2 for 1 and n
end
This should result in less work than what you are doing. But for large numbers this is a hard task to achieve.

Answer 4

If you want to stick with your algorithm, here is an optimization.
def div(num)
oddity = false
(1..num).each{|x| if (num % x == 0) then oddity = !oddity end}
oddity ? "odd" : "even"
end