五位数\(16807=7^5\)也是一个五次幂,同样的,九位数\(134217728=8^9\)也是一个九次幂.求有多少个\(n\)位正整数同时也是\(n\)次幂? 分析:设题目要求的幂的底为\(n\),指数为\(k\),则这个幂应为\(k\)位数,则有: \[ 10^{k-1}<n^k<10^k \Rightarrow k-1<k\cdot log_{10}n<k \] 因为\(k\ge1\),则对于不等式\(k\cdot log_{10}n<k\),有\(log_{10}…

The 5-digit number, 16807=75, is also a fifth power. Similarly, the 9-digit number, 134217728=89, is a ninth power. How many n-digit positive integers exist which are also an nth power? 这种数字满足下面条件: 对于数位为x的数S=k^x 有 10^(x-1)<=k^x<=10^x-1 #include &quo…

一个古戈尔也就是\(10^{100}\)是一个天文数字,一后面跟着一百个零.\(100^{100}\)更是难以想像的大,一后面跟着两百个零.但是尽管这个数字很大,它们各位数字的和却只等于一.考虑两个自然数\(a,b\)形成的指数\(a^b(a,b<100)\),其最大的各位数字之和是多少? 分析:此题思路比较直接,暴力求解即可.在题目给它的范围内,\(a^b\)可以形成9801个数,计算这些数,求其各位数之和并返回其最大值,即为题目所求. # time cost = 190 ms ± 515 µ…

题目链接 The 5-digit number, 16807=75, is also a fifth power. Similarly, the 9-digit number, 134217728=89, is a ninth power. How many n-digit positive integers exist which are also an nth power? 这个题目有点坑: 先说自己的思路<虽然方法不是很好> 根据题意可知道: a的b次方 除以 最小的b位数(如:1,10…

A number chain is created by continuously adding the square of the digits in a number to form a new number until it has been seen before. For example, 44 → 32 → 13 → 10 → 1 → 1 85 → 89 → 145 → 42 → 20 → 4 → 16 → 37 → 58 → 89 Therefore any chain that…

题意:求出100!的各位数字和. /************************************************************************* > File Name: euler020.c > Author: WArobot > Blog: http://www.cnblogs.com/WArobot/ > Created Time: 2017年06月28日 星期三 11时46分46秒 ***************************…

题意: 215 = 32768,而32768的各位数字之和是 3 + 2 + 7 + 6 + 8 = 26. 21000的各位数字之和是多少? 思路:大数乘法,计算 210 × 100 可加速计算,每次超过1000进位 /************************************************************************* > File Name: euler016.c > Author: WArobot > Blog: http://www.…

通过替换*3这样一个两位数的第一位,我们可以发现形成的九个数字有六个是质数,即13, 23,43,53,73,83.类似的,如果我们用同样的数字替换56**3这样一个五位数的第三位和第四位,会生成56003, 56113, 56333, 56443, 56663, 56773, 56993七个质数,事实是56003是拥有这个性质的最小质数.已知对于一个质数,可以通过使用相同的数字替换这个质数的一部分(不一定是相邻的数位),可以生成八个质数,求满足这个条件的最小质数. 分析:欧拉工程的前五十题都是…

直接python搞过.没啥好办法.看了下别人做的,多数也是大数乘法搞过. 如果用大数做的话,c++写的话,fft优化大数乘法,然后快速幂一下就好了.…

本题来自 Project Euler 第17题:https://projecteuler.net/problem=17 ''' Project Euler 17: Number letter counts If the numbers 1 to 5 are written out in words: one, two, three, four, five, then there are 3 + 3 + 5 + 4 + 4 = 19 letters used in total. If all th…

本题来自 Project Euler 第16题:https://projecteuler.net/problem=16 ''' Project Euler 16: Power digit sum 2**15 = 32768 and the sum of its digits is 3 + 2 + 7 + 6 + 8 = 26. What is the sum of the digits of the number 2**1000? Answer: 1366 ''' print(sum(int(i…

In Problem 42 we dealt with triangular problems, in Problem 44 of Project Euler we deal with pentagonal number, I can only wonder if we have to deal with septagonal numbers in Problem 46. Anyway the problem reads Pentagonal numbers are generated by t…

本题来自 Project Euler 第20题:https://projecteuler.net/problem=20 ''' Project Euler: Problem 20: Factorial digit sum n! means n × (n − 1) × ... × 3 × 2 × 1 For example, 10! = 10 × 9 × ... × 3 × 2 × 1 = 3628800, and the sum of the digits in the number 10! i…

本题来自 Project Euler 第11题:https://projecteuler.net/problem=11 # Project Euler: Problem 10: Largest product in a grid # In the 20×20 grid below, four numbers along a diagonal line have been marked in red. # The product of these numbers is 26 × 63 × 78 ×…

上一次接触 project euler 还是2011年的事情,做了前三道题,后来被第四题卡住了,前面几题的代码也没有保留下来. 今天试着暴力破解了一下,代码如下: (我大概是第 172,719 个解出这道题的人) program 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.…

开始做 Project Euler 的练习题.网站上总共有565题,真是个大题库啊! # Project Euler, 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…

题意:三个正整数a + b + c = 1000,a*a + b*b = c*c.求a*b*c. 解法:可以暴力枚举,但是也有数学方法. 首先,a,b,c中肯定有至少一个为偶数,否则和不可能为以上两个等式均不会成立.然后,不可能a,b为奇c为偶,否则a*a%4=1, b*b%4=1, 有(a*a+b*b) %4 = 2,而c*c%4 = 0.也就是说,a和b中至少有一个偶数. 这是勾股数的一个性质,a,b中至少有一个偶数. 然后,解决过程见下(来自project euler的讨论): tag:m…

project euler 169 题目链接:https://projecteuler.net/problem=169 参考题解:http://tieba.baidu.com/p/2738022069 #include<bits/stdc++.h> using namespace std; #define fi first #define se second #define mp make_pair #define pb push_back #define rep(i, a, b) for(i…

题目要求是: The four adjacent digits in the 1000-digit number that have the greatest product are 9 × 9 × 8 × 9 = 5832. 7316717653133062491922511967442657474235534919493496983520312774506326239578318016984801869478851843858615607891129494954595017379583319…

Project Euler: 欧拉计划是一系列挑战数学或者计算机编程问题,解决这些问题需要的不仅仅是数学功底. 启动这一项目的目的在于,为乐于探索的人提供一个钻研其他领域并且学习新知识的平台,将这一平台打造一个有趣和休闲 的环境. 项目主页:https://projecteuler.net 第一题 Multiples of 3 and 5 If we list all the natural numbers below 10 that are multiples of 3 or 5, we ge…

本题来自 Project Euler 第22题:https://projecteuler.net/problem=22 ''' Project Euler: Problem 22: Names scores Using names.txt (right click and 'Save Link/Target As...'), a 46K text file containing over five-thousand first names, begin by sorting it into al…

本题来自 Project Euler 第21题:https://projecteuler.net/problem=21 ''' Project Euler: Problem 21: Amicable numbers Let d(n) be defined as the sum of proper divisors of n (numbers less than n which divide evenly into n). If d(a) = b and d(b) = a, where a ≠ b…

本题来自 Project Euler 第19题:https://projecteuler.net/problem=19 ''' How many Sundays fell on the first of the month during the twentieth century (1 Jan 1901 to 31 Dec 2000)? Answer: 171 ''' from datetime import * firstDay = date(1901,1,1) lastDay = date(…

本题来自 Project Euler 第15题:https://projecteuler.net/problem=15 ''' Project Euler: Problem 15: Lattice paths Starting in the top left corner of a 2×2 grid, and only being able to move to the right and down, there are exactly 6 routes to the bottom right…

本题来自 Project Euler 第14题:https://projecteuler.net/problem=14 ''' Project Euler: Problem 14: Longest Collatz sequence The following iterative sequence is defined for the set of positive integers: n → n/2 (n is even) n → 3n + 1 (n is odd) Using the rule…

本题来自 Project Euler 第13题:https://projecteuler.net/problem=13 # Project Euler: Problem 13: Large sum # Work out the first ten digits of the sum of the following one-hundred 50-digit numbers. # Answer: 5537376230 numbers = '''371072875339021027987979982…

本题来自 Project Euler 第12题:https://projecteuler.net/problem=12 # Project Euler: Problem 12: Highly divisible triangular number # The sequence of triangle numbers is generated by adding the natural numbers. # So the 7th triangle number would be 1 + 2 + 3…

本题来自 Project Euler 第10题:https://projecteuler.net/problem=10 # Project Euler: 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. # Answer: 142913828922 def f(x):…

本题来自 Project Euler 第9题:https://projecteuler.net/problem=9 # Project Euler: 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, 3**2 + 4**2 = 9 +…

本题来自 Project Euler 第8题:https://projecteuler.net/problem=8 # Project Euler: 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. # Find the thirteen adjac…