By listing the first six prime numbers: 2, 3, 5, 7, 11, and 13, we can see that the 6th prime is 13.

What is the 10 001st prime number?

http://projecteuler.net/problem=7

highlight below for my solution:

#using a prime number set datastructure - https://gist.github.com/aausch/6709819
p =  PrimeSet()

i = 10001

while (len(p)<10001):
    i in p
    i += i

print sorted(list(p))[10000]

(see also my solution to problem #5)

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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?

http://projecteuler.net/problem=5

highlight below for my solution:

#using a prime number set datastructure - https://gist.github.com/aausch/6709819
p = PrimeSet()

def min_product(n):
    n in p #initialize the PrimeSet with all primes less than n
    product = 1
    for prime in p:
        product = product * (prime ** (int(n ** (1.0/prime))))
    return product

print min_product(20)

(see also my solution to problem #3)

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

What is the largest prime factor of the number 600851475143 ?

http://projecteuler.net/problem=3

highlight below for my solution:

#using a prime number set datastructure - https://gist.github.com/aausch/6709819
p_set = PrimeSet()
n = 600851475143
sqrt = int(n ** 0.5) 
p_set[sqrt]
max_factor = 1
for x in p_set:
    if n % x == 0 and x > max_factor:
        max_factor = x       
print max_factor

[More programming riddles]