欧拉计划 Project Euler 1-7

Project Euler 1-7

Project Euler: https://projecteuler.net/
Project Euler | 欧拉计划: https://pe-cn.github.io/
Project Euler 1-7
Project Euler 8-14
Project Euler 15-21 & 67
Project Euler 22-28
Project Euler 29-35
Project Euler 36-42
Project Euler 43-49
Project Euler 50-56
Project Euler 57-63
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随缘更新

Problem 1

Find the sum of the multiples of 3 or 5 below 1000

• sum(the multiples of 3 below 1000) + sum(the multiples of 5 below 1000) - sum(the multiples of 3*5 below 1000)

num_3 = 1000//3
num_5 = 1000//5 - 1
num_3x5 = 1000//(3*5)
Sum_3 = (3+num_3*3)*num_3/2
Sum_5 = (5+num_5*5)*num_5/2
Sum_3x5 = (3*5+num_3x5*3*5)*num_3x5/2
Sum = Sum_3 + Sum_5 - Sum_3x5
print(Sum)

233168.0

Problem 2

Even Fibonacci numbers

• 有个好用的python语法

  a, b = b, a+b

Sum = 0
num_0 = 1
num_1 = 2
while num_1 < 4000000:
    #1, 2, 3, 5, 8, 13, 21, 34, 55, 89, ...
    if num_1%2 == 0:
        Sum += num_1
    num_0, num_1 = num_1, num_0+num_1
#     num_temp = num_1
#     num_1 += num_0
#     num_0 = num_temp
print(Sum)

4613732

Problem 3: Largest prime factor(质因子)

What is the largest prime factor of the number 600851475143 ?

• 通过NUMBER除以自身的因数，逐步缩小NUMBER的规模

NUMBER = 600851475143
i = 2
res = 1
while NUMBER>2:
    if NUMBER%i == 0:
        NUMBER = NUMBER//i
        res = i
    else:
        i+=1
print(res)

6857

Problem 4: Largest palindrome(回文数) product

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

• 暴力搜索：对900个6位回文数由大到小一次进行因式分解，查找是否存在3位x3位的组合
• python语法：在Python中的while或者for循环之后还可以有else子句，作用是for循环中if条件一直不满足，则最后就执行else语句

# 999 999
# 998 899
# ...
for i in range(999, 100, -1):
    num_temp = i%10*100 + (i//10)%10*10 + i//100
    num = i * 1000 + num_temp
    for i in range(999, 100, -1):
        if num%i == 0:
            if num//i <= 999 and num//i >= 100:
                print(i, num//i, num)
                break
    else:
        continue
    break

993 913 906609

Problem 5: Smallest multiple

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

• 遍历1-20，找出每个数的因数且从中去除之前记录下的因数

n = 20
ans = 1
L = []
for i in range(1,n+1):
    for j in L:
        if i%j == 0:
            i = i//j
    L.append(i)
print(L)
for i in L:
    ans *= i
print(ans)

[1, 2, 3, 2, 5, 1, 7, 2, 3, 1, 11, 1, 13, 1, 1, 2, 17, 1, 19, 1]
232792560

Problem 6: Sum square difference

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

• 平方和公式:$$\sum _{i=1}^n{n}^2=\frac{n(n+1)(2n+1)}{6}$$

n = 100
# def sum_of_squares(n):
#     if n==1:
#         return 1
#     else:
#         return sum_of_squares(n-1) + n**2
# ans = ((1+n)*n//2)**2 - sum_of_squares(n)
ans = ((1+n)*n//2)**2 - n*(n+1)*(2*n+1)//6
print(ans)

25164150

Problem 7: 10001st prime

What is the 10 001st prime number?

• 因数都是成对出现，如果出现一个大于$\sqrt{x}$的因数，必然有一个小于$\sqrt{x}$的因数存在，因此遍历到$\sqrt{x}$就可以判定一个数是不是质数，缩短判断质数的时间

import time
t = time.clock()
i=0
k=2
n=10001
while i<n-1:
    k+=1
#     for j in range(2, k):
#         if k%j == 0:
#             break
    for j in range(2,int(k**0.5)+1):
        if k%j==0:
            break
    else:
        i+=1 
print(k)
# print(time.clock() - t)

104743