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Project Euler, Problems 1-10





In [34]:



    


using TimeIt
using BenchmarkTools



    


















    






INFO: Precompiling module JLD.
WARNING: could not import Base.lastidx into LegacyStrings

Problem 1

If we list all the natural numbers below 10 that are multiples of 3 or 5, we get 3, 5, 6 and 9. The sum of these multiples is 23. Find the sum of all the multiples of 3 or 5 below 1000.





In [11]:



    


@time sum([x for x in 1:999 if ((x%3 ==0) || (x%5 == 0))])



    


















    






  0.056626 seconds (28.83 k allocations: 1.264 MB)










    
Out[11]:








233168

















In [12]:



    


@time sum(x for x in 1:999 if ((x%3 == 0) || (x%5 ==0)))



    


















    






  0.058550 seconds (30.46 k allocations: 1.302 MB)










    
Out[12]:








233168

Problem 2

Each new term in the Fibonacci sequence is generated by adding the previous two terms. By starting with 1 and 2, the first 10 terms will be:

1, 2, 3, 5, 8, 13, 21, 34, 55, 89, ...

By considering the terms in the Fibonacci sequence whose values do not exceed four million, find the sum of the even-valued terms.





In [1]:



    


immutable Fibonacci
end



    











In [2]:



    


Base.start(f::Fibonacci) = (0,1)
Base.next(f::Fibonacci, state) = (state[2], (state[2], state[1] + state[2]))
Base.done(f::Fibonacci, state) = false 
Base.iteratorsize(f::Fibonacci) = Base.IsInfinite()



    











In [13]:



    


sum(collect(filter(x -> iseven(x) && x < Int(4e6), take(Fibonacci(), 50))))



    


















    
Out[13]:








4613732

















In [35]:



    


using Lazy



    











In [18]:



    


@timeit sum(filter(iseven, takewhile(x->x<4e6,Fibonacci())))



    


















    






1000 loops, best of 3: 1.08 ms per loop










    
Out[18]:








0.00107524478

Problem 3

The prime factors of 13195 are 5, 7, 13 and 29.

What is the largest prime factor of the number 600851475143 ?





In [49]:



    


#sieve of Erasthosenes
function PrimeSieve(N::Int64)
    isprime = trues(N)
    for i = 2:Int64(trunc(sqrt(N)))
        if isprime[i]
            for j in (i*i):i:N
                isprime[j] = false
            end
        end
    end
    return filter(x -> isprime[x], 2:N)
end



    


















    
Out[49]:








PrimeSieve (generic function with 1 method)

















In [20]:



    


function LargestPrimeFactor(N::Int64)
    for x in reverse(Primes(10000))
        if N % x == 0
            return x
        end
    end
    return N
end



    


















    






WARNING: Method definition LargestPrimeFactor(Int64) in module Main at In[2]:2 overwritten









    
Out[20]:








LargestPrimeFactor (generic function with 1 method)










    






 at In[20]:2.

















In [23]:



    


@timeit LargestPrimeFactor(600851475143)



    


















    






1000 loops, best of 3: 182.20 µs per loop










    
Out[23]:








0.000182200056

Problem 4

A palindromic number reads the same both ways. The largest palindrome made from the product of two 2-digit numbers is 9009 = 91 × 99.

Find the largest palindrome made from the product of two 3-digit numbers.





In [24]:



    


@timeit maximum(filter(x -> reverse("$x") == "$x", [x*y for x in 100:999 for y in x:999]))



    


















    






1 loops, best of 3: 75.79 ms per loop










    
Out[24]:








0.075786782

















In [27]:



    


@timeit maximum(filter(x -> reverse("$x") == "$x", (x*y for x in 999:-1:100 for y in 999:-1:x)))



    


















    






1 loops, best of 3: 85.85 ms per loop










    
Out[27]:








0.085848368

















In [35]:



    


@timeit maximum(unique([x*y for x in 100:999 for y in x:999 if reverse("$(x*y)") == "$(x*y)"]))



    


















    






1 loops, best of 3: 83.40 ms per loop










    
Out[35]:








0.083397317

Problem 5

2520 is the smallest number that can be divided by each of the numbers from 1 to 10 without any remainder.

What is the smallest positive number that is evenly divisible by all of the numbers from 1 to 20?





In [2]:



    


#Euclid's algorithm for GCD
function GCD(a::Int64, b::Int64)
    if b == 0
        return a
    else
        return GCD(b, a % b)
    end
end 
LCM(a::Int64, b::Int64) = div(abs(a), GCD(a,b)) * abs(b)



    


















    
Out[2]:








LCM (generic function with 1 method)

















In [64]:



    


@timeit reduce(LCM, 1, 1:20)



    


















    






1000000 loops, best of 3: 948.38 ns per loop










    
Out[64]:








9.48378315e-7

Problem 6

Find the difference between the sum of the squares of the first one hundred natural numbers and the square of the sum.





In [71]:



    


sum(1:100)^2 - sum(map(x -> x^2, 1:100))



    


















    
Out[71]:








25164150

Problem 7

What is the 10001st prime number?





In [1]:



    


immutable PrimeIt
end



    











In [30]:



    


Base.start(::PrimeIt) = (2, Dict{Int64, Int64}())
function Base.next(::PrimeIt, state)
    q = state[1]
    D = state[2]
    while true
        if q not in keys(D)
            D[q * q] = q
            return q, (q+1, D)
        else
            for p in D[q]
                if !(p+q in keys(D))
                    D[p+q] = []
                end
                push!(D[p+q], p)
            end
            delete!(D, q)
        end
    end
end
Base.done(::PrimeIt, q) = false
Base.iteratorsize(::PrimeIt) = Base.IsInfinite()



    


















    






WARNING: Method definition start(Main.PrimeIt) in module Main at In[9]:2 overwritten at In[30]:1.
WARNING: Method definition next(Main.PrimeIt, Any) in module Main at In[9]:4 overwritten at In[30]:3.
WARNING: Method definition done(Main.PrimeIt, Any) in module Main at In[9]:12 overwritten at In[30]:20.
WARNING: Method definition iteratorsize(Main.PrimeIt) in module Main at In[9]:13 overwritten at In[30]:21.

















In [41]:



    


@timeit last(collect(take(PrimeIt(), 10001)))



    


















    






10 loops, best of 3: 17.51 ms per loop










    
Out[41]:








0.017505281

Problem 8





In [8]:



    


bigString = replace("""73167176531330624919225119674426574742355349194934
96983520312774506326239578318016984801869478851843
85861560789112949495459501737958331952853208805511
12540698747158523863050715693290963295227443043557
66896648950445244523161731856403098711121722383113
62229893423380308135336276614282806444486645238749
30358907296290491560440772390713810515859307960866
70172427121883998797908792274921901699720888093776
65727333001053367881220235421809751254540594752243
52584907711670556013604839586446706324415722155397
53697817977846174064955149290862569321978468622482
83972241375657056057490261407972968652414535100474
82166370484403199890008895243450658541227588666881
16427171479924442928230863465674813919123162824586
17866458359124566529476545682848912883142607690042
24219022671055626321111109370544217506941658960408
07198403850962455444362981230987879927244284909188
84580156166097919133875499200524063689912560717606
05886116467109405077541002256983155200055935729725
71636269561882670428252483600823257530420752963450""", '\n', "")



    


















    
Out[8]:








"7316717653133062491922511967442657474235534919493496983520312774506326239578318016984801869478851843858615607891129494954595017379583319528532088055111254069874715852386305071569329096329522744304355766896648950445244523161731856403098711121722383113622298934233803081353362766142828064444866452387493035890729629049156044077239071381051585930796086670172427121883998797908792274921901699720888093776657273330010533678812202354218097512545405947522435258490771167055601360483958644670632441572215539753697817977846174064955149290862569321978468622482839722413756570560574902614079729686524145351004748216637048440319989000889524345065854122758866688116427171479924442928230863465674813919123162824586178664583591245665294765456828489128831426076900422421902267105562632111110937054421750694165896040807198403850962455444362981230987879927244284909188845801561660979191338754992005240636899125607176060588611646710940507754100225698315520005593572972571636269561882670428252483600823257530420752963450"

















In [35]:



    


@benchmark maximum([mapreduce(x->parse(Int64,x), *, 1, y) for y in [bigString[i:i+12] for i in 1:length(bigString)-12] if !('0' in y)])



    


















    
Out[35]:








BenchmarkTools.Trial: 
  memory estimate:  172.06 KiB
  allocs estimate:  3514
  --------------
  minimum time:     228.361 μs (0.00% GC)
  median time:      239.769 μs (0.00% GC)
  mean time:        271.703 μs (7.85% GC)
  maximum time:     4.371 ms (88.65% GC)
  --------------
  samples:          10000
  evals/sample:     1
  time tolerance:   5.00%
  memory tolerance: 1.00%

Problem 9





In [48]:



    


[a*b*c for a in 1:1000 for b in a:1000 for c in b:1000 if (a^2 + b^2 == c^2 && a+b+c == 1000)]



    


















    
Out[48]:








1-element Array{Int64,1}:
 31875000

Problem 10





In [51]:



    


@benchmark sum(PrimeSieve(2000000))



    


















    
Out[51]:








BenchmarkTools.Trial: 
  memory estimate:  3.28 MiB
  allocs estimate:  18
  --------------
  minimum time:     30.390 ms (0.00% GC)
  median time:      31.466 ms (0.00% GC)
  mean time:        32.183 ms (1.03% GC)
  maximum time:     42.452 ms (5.69% GC)
  --------------
  samples:          156
  evals/sample:     1
  time tolerance:   5.00%
  memory tolerance: 1.00%

















In [52]:



    






    


















    
Out[52]:








66666.66666666667

















In [ ]:

Content source: gribeill/ProjectEuler

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