# Project Euler in Python Series 1

## Problem 1 Multiples of 3 and 5

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.

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def problem_1():
sum = 0
for x in range(1000):
if x % 15 == 0:
sum += x
elif x % 5 == 0:
sum += x
elif x % 3 == 0:
sum += x
return sum
print(problem_1())

## Problem 2 Even Fibonacci numbers

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.

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def problem_2():
fibonacci = [1, 1]
while fibonacci[-1] <= 4000000:
fibonacci.append(fibonacci[-1] + fibonacci[-2])
sum = 0
for x in fibonacci:
if x % 2 == 0:
sum += x
return sum
print(problem_2())

## Problem 3 Largest prime factor

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

What is the largest prime factor of the number 600851475143 ?

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def smallest_prime_factor(x):
for y in range(2, x):
if x % y == 0:
return y
return x
def problem_3():
num = 600851475143
while num != smallest_prime_factor(num):
f = smallest_prime_factor(num)
if f <= num:
num //= f
else:
break
return num
print(problem_3())

## Problem 4 Largest palindrome product

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.

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def problem_4():
list = []
for x in range(999, 100, -1):
for y in range(999, 100, -1):
l = str(x*y)
reversed = [l[i] for i in range(len(l) - 1, -1, -1)]
if str(x * y) == "".join(reversed):
list.append(x * y)
return max(list)
print(problem_4())

## Problem 5 Smallest Multiple

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?

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def gcd(x,y):
r = 1
while r != 0:
if x > y:
x,y = y,x
r = y % x
if r == 0:
return x
else:
y,x = x,r
print(gcd(12,6))
def lcm(x, y):
return x * y // gcd(x, y)
def problem_5():
x = 1
for y in range(1, 21):
x = lcm(x, y)
return x
print(problem_5())

## Problem 6 Sum square difference

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.

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def problem_6():
sums = 0
for x in range(1, 101):
sums += x
sums = sums * sums
squares = 0
square = [x*x for x in range(1,101)]
for z in square:
squares += z
return sums - squares
print(problem_6())

## Problem 8 Largest product in a series

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?

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a = """73167176531330624919225119674426574742355349194934
96983520312774506326239578318016984801869478851843
85861560789112949495459501737958331952853208805511
12540698747158523863050715693290963295227443043557
66896648950445244523161731856403098711121722383113
62229893423380308135336276614282806444486645238749
30358907296290491560440772390713810515859307960866
70172427121883998797908792274921901699720888093776
65727333001053367881220235421809751254540594752243
52584907711670556013604839586446706324415722155397
53697817977846174064955149290862569321978468622482
83972241375657056057490261407972968652414535100474
82166370484403199890008895243450658541227588666881
16427171479924442928230863465674813919123162824586
17866458359124566529476545682848912883142607690042
24219022671055626321111109370544217506941658960408
07198403850962455444362981230987879927244284909188
84580156166097919133875499200524063689912560717606
05886116467109405077541002256983155200055935729725
71636269561882670428252483600823257530420752963450"""
def problem_8():
global a
a = a.replace("\n", "")
max = 1
for x in range(len(a) - 13):
product = 1
for y in range(13):
product *= int(a[x + y])
if product > max:
max = product
return max
print((problem_8()))

## Problem 9 Special Pythagorean triplet

A Pythagorean triplet is a set of three natural numbers, a < b < c, for which,

\[a^2 + b^2 = c^2\]For example, 32 + 42 = 9 + 16 = 25 = 52.

There exists exactly one Pythagorean triplet for which a + b + c = 1000. Find the product abc.

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def problem_9():
for x in range(1, 1000):
for y in range(x + 1, 1000):
for z in range(y + 1,1000):
if x < y < z:
if x + y + z == 1000:
if x**2 + y**2 == z**2:
return x*y*z
print(problem_9())

## Problem 10 Summation of primes

The sum of the primes below 10 is 2 + 3 + 5 + 7 = 17.

Find the sum of all the primes below two million.

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import sympy
def problem_10():
sum = 0
for x in range(3, 2000000, 2):
if sympy.isprime(x):
sum += x
return sum + 2
print(problem_10())