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Consider the fraction,n/d, wherenanddare positive integers. Ifndand HCF(n,d)=1, it is called a reduced proper fraction.If we list the set of reduced proper fractions ford8 in ascending order of size, we get:1/8, 1/7, 1/6, 1/5, 1/4, 2/7, 1/3,3/8, 2/5, 3/7, 1/2, 4/7, 3/5, 5/8, 2/3, 5/7, 3/4, 4/5, 5/6, 阅读全文
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Thenthterm of the sequence of triangle numbers is given by,tn= ½n(n+1); so the first ten triangle numbers are:1, 3, 6, 10, 15, 21, 28, 36, 45, 55, ...By converting each letter in a word to a number corresponding to its alphabetical position and adding these values we form a word value. For exam 阅读全文
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We shall say that ann-digit number is pandigital if it makes use of all the digits 1 tonexactly once. For example, 2143 is a 4-digit pandigital and is also prime.What is the largestn-digit pandigital prime that exists?题目大意：如果一个数字将1到n的每个数字都使用且只使用了一次，我们将其称其为一个n位的pandigital数。例如，2143是一个4位的pandigital数，并且 阅读全文
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Euler's Totient function, φ(n) [sometimes called the phi function], is used to determine the number of positive numbers less than or equal tonwhich are relatively prime ton. For example, as 1, 2, 4, 5, 7, and 8, are all less than nine and relatively prime to nine, φ(9)=6. The number 1 is conside 阅读全文
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The number 145 is well known for the property that the sum of the factorial of its digits is equal to 145:1! + 4! + 5! = 1 + 24 + 120 = 145Perhaps less well known is 169, in that it produces the longest chain of numbers that link back to 169; it turns out that there are only three such loops that ex 阅读全文
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It was proposed by Christian Goldbach that every odd composite number can be written as the sum of a prime and twice a square.9 = 7 + 21215 = 7 + 22221 = 3 + 23225 = 7 + 23227 = 19 + 22233 = 31 + 212It turns out that the conjecture was false.What is the smallest odd composite that cannot be written 阅读全文
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The cube, 41063625 (3453), can be permuted to produce two other cubes: 56623104 (3843) and 66430125 (4053). In fact, 41063625 is the smallest cube which has exactly three permutations of its digits which are also cube.Find the smallest cube for which exactly five permutations of its digits are cube. 阅读全文
摘要：
Consider the fraction,n/d, wherenanddare positive integers. Ifndand HCF(n,d)=1, it is called a reduced proper fraction.If we list the set of reduced proper fractions ford8 in ascending order of size, we get:1/8, 1/7, 1/6, 1/5, 1/4, 2/7, 1/3, 3/8, 2/5, 3/7, 1/2, 4/7, 3/5, 5/8, 2/3, 5/7, 3/4, 4/5, 5/6 阅读全文
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There are exactly ten ways of selecting three from five, 12345:123, 124, 125, 134, 135, 145, 234, 235, 245, and 345In combinatorics, we use the notation,5C3= 10.In general,It is not untiln= 23, that a value exceeds one-million:23C10= 1144066.How many, not necessarily distinct, values of nCr, for 1n1 阅读全文
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The arithmetic sequence, 1487, 4817, 8147, in which each of the terms increases by 3330, is unusual in two ways: (i) each of the three terms are prime, and, (ii) each of the 4-digit numbers are permutations of one another.There are no arithmetic sequences made up of three 1-, 2-, or 3-digit primes, 阅读全文
