In [1]:
solution = 0
N = 0
while N < 1000:
if (N % 3 == 0) or (N % 5 == 0):
solution = solution + N
N+=1
print solution
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 [7]:
s=0
a=1
b=1
while b < 4E6:
if b % 2 ==0: s=s+b;
c=a+b
a=b
b=c
print s
In [31]:
n=600851475143 #13195
a=2
while not a > n:
if n%a:
a+=1
else:
n//=a
print a
In [8]:
a=999
b=999
while a>100 and b>100:
c = a*b
if str(c) == str(c)[::-1]:
print a,"*",b,"=",c
break
a=a-1
b=b-1
In [31]:
#brute force
n=2520
l=n*11*12*13*14*15*16*17*18*19
while n<l:
reminder=[n % (i+1) for i in range(20)]
if sum(reminder) == 0:
print n
break
n=n+1
In [80]:
# 1*2*3*4*5*7*3=2520
# prime number: must be a factor
# composite number: factorize and remove smaller prime number
def is_prime(n):
if n < 1:
return False
elif n < 4:
return True
elif (n % 2 == 0) or (n % 3 == 0):
return False
i = 5
while i*i < n + 1:
if (n % i == 0):
return false
i=i+1
#while i*i < n + 1:
# if (n % i == 0) or (n % (i + 2) == 0):
# return False
# i = i + 6
return True
def prime_factors(n):
i = 2
factors = []
while i * i <= n:
if n % i:
i += 1
else:
n //= i
factors.append(i)
if n > 1:
factors.append(n)
return factors
fac=[]
for i in range(20):
if is_prime(i+1):
fac.append(i+1)
else:
test=fac[:]
for p in prime_factors(i+1):
if p in test:
test.remove(p)
else:
fac.append(p)
print fac
product = 1
for x in fac:
product *= x
print product
The sum of the squares of the first ten natural numbers is,
$1^2 + 2^2 + ... + 10^2 = 385$
The square of the sum of the first ten natural numbers is,
$(1 + 2 + ... + 10)^2 = 55^2 = 3025$
Hence the difference between the sum of the squares of the first ten natural numbers and the square of the sum is 3025 − 385 = 2640.
Find the difference between the sum of the squares of the first one hundred natural numbers and the square of the sum.
In [85]:
sum_of_the_square = sum([i*i for i in range(1,100+1)])
square_of_the_sum = sum(range(1,100+1))
square_of_the_sum = square_of_the_sum*square_of_the_sum
print square_of_the_sum - sum_of_the_square
In [91]:
def is_prime(n):
if n < 2:
return False
elif n < 4:
return True
elif (n % 2 == 0) or (n % 3 == 0):
return False
i = 5
while i*i < n + 1:
if (n % i == 0):
return False
i=i+1
return True
i=1
count_prime=0
while count_prime<10001:
i+=1
if is_prime(i):
count_prime+=1
print i,"#",count_prime
The four adjacent digits in the 1000-digit number that have the greatest product are 9 × 9 × 8 × 9 = 5832.
73167176531330624919225119674426574742355349194934 96983520312774506326239578318016984801869478851843 85861560789112949495459501737958331952853208805511 12540698747158523863050715693290963295227443043557 66896648950445244523161731856403098711121722383113 62229893423380308135336276614282806444486645238749 30358907296290491560440772390713810515859307960866 70172427121883998797908792274921901699720888093776 65727333001053367881220235421809751254540594752243 52584907711670556013604839586446706324415722155397 53697817977846174064955149290862569321978468622482 83972241375657056057490261407972968652414535100474 82166370484403199890008895243450658541227588666881 16427171479924442928230863465674813919123162824586 17866458359124566529476545682848912883142607690042 24219022671055626321111109370544217506941658960408 07198403850962455444362981230987879927244284909188 84580156166097919133875499200524063689912560717606 05886116467109405077541002256983155200055935729725 71636269561882670428252483600823257530420752963450
Find the thirteen adjacent digits in the 1000-digit number that have the greatest product. What is the value of this product?
In [115]:
s='''73167176531330624919225119674426574742355349194934
96983520312774506326239578318016984801869478851843
85861560789112949495459501737958331952853208805511
12540698747158523863050715693290963295227443043557
66896648950445244523161731856403098711121722383113
62229893423380308135336276614282806444486645238749
30358907296290491560440772390713810515859307960866
70172427121883998797908792274921901699720888093776
65727333001053367881220235421809751254540594752243
52584907711670556013604839586446706324415722155397
53697817977846174064955149290862569321978468622482
83972241375657056057490261407972968652414535100474
82166370484403199890008895243450658541227588666881
16427171479924442928230863465674813919123162824586
17866458359124566529476545682848912883142607690042
24219022671055626321111109370544217506941658960408
07198403850962455444362981230987879927244284909188
84580156166097919133875499200524063689912560717606
05886116467109405077541002256983155200055935729725
71636269561882670428252483600823257530420752963450'''
import re
s=re.sub("[^0-9]", "", s)
#ss = [s[i:i+13] for i in range(len(s)-12)]
#print ss[-1]
maxproduct=1
for i in range(len(s)-12):
ss = map(int,s[i:i+13])
product = 1
for x in ss:
product *= x
if product > maxproduct:
maxproduct=product
print maxproduct
In [132]:
# By substituting c=1000-(a+b) into Pythagorean equality, we get
# n^2 = 2n(a+b) - 2ab, n=1000 here
def f(a,b,n):
return 2*n*(a+b) - 2*a*b - n*n
import numpy as np
n=1000
X,Y=np.meshgrid(range(1,n),range(1,n))
X=np.tril(X,-1).ravel()
Y=np.tril(Y,-1).ravel()
for p in zip(X[X>0],Y[Y>0]):
if f(p[0],p[1],n) == 0:
print p[0],p[1], n-p[0]-p[1]
print p[0]*p[1]*(n-p[0]-p[1])
In [154]:
def is_prime(n):
if n < 2:
return False
elif n < 4:
return True
elif (n % 2 == 0) or (n % 3 == 0):
return False
i = 5
while i*i < n + 1:
if (n % i == 0) or (n % (i + 2) == 0):
return False
i = i + 6
return True
i=1
s=0
while i<2e6:
i+=1
if is_prime(i):
s+=i
print i,"SUM",s
In [ ]: