Sum square difference

Problem 6 

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.

Answer 25164150

Code

num = 100

temp = 0

diff = 0

temp =  num ** 2

diff = (3 * temp ** 2 + 2 * temp * num - 3 * temp -2 * num )/12

print(diff)

Smallest multiple

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?

Answer 232792560

Code

def gcd(a, b):

    while b != 0:

        a, b = b, a % b

    return a

def lcm(a, b):

    return a*b/gcd(a, b)

lcmNum = 1

for i in range(2,21):

    lcmNum = lcm(lcmNum,i)

print(lcmNum)

Largest palindrome product

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.

Answer

number = 906609

Code

print(max([x*y for x in range(999,900,-1) for y in range(x,900,-1) if str(x*y) == str(x*y)[::-1]]))

Largest prime factor
Problem 3
The prime factors of 13195 are 5, 7, 13 and 29.
What is the largest prime factor of the number 600851475143 ?

Answer
number = 600851475143

i = 2
m = 2
while number != 1:
    if number % i == 0:
        number = number / i
        m = i
    i = i + 1

print m