Problem Euler’s Totient function, ?(n) [sometimes called the phi function], is used to determine the number of numbers less than n which are relatively prime to n. For example, as 1, 2, 4, 5, 7, and 8, are all less than nine and relatively prime to nine, ?(9)=6. It can be seen that n=6 produces …
Problem Consider the fraction, n/d, where n and d are positive integers. If n<d and HCF(n,d)=1, it is called a reduced proper fraction. If we list the set of reduced proper fractions for d <= 8 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, …
Problem Take the number 192 and multiply it by each of 1, 2, and 3: 192 x 1 = 192 192 x 2 = 384 192 x 3 = 576 By concatenating each product we get the 1 to 9 pandigital, 192384576. We will call 192384576 the concatenated product of 192 and (1,2,3) The same …
Problem 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, …
Problem Find the unique positive integer whose square has the form 1_2_3_4_5_6_7_8_9_0, where each “_” is a single digit. Solution Initially I started solving this problem by creating all permutations of 19 digits numbers which fits the required pattern and iteratively square rooting them to find a square root which is a whole number. Unfortunately …
Problem The fraction 49/98 is a curious fraction, as an inexperienced mathematician in attempting to simplify it may incorrectly believe that 49/98 = 4/8, which is correct, is obtained by cancelling the 9s. We shall consider fractions like, 30/50 = 3/5, to be trivial examples. There are exactly four non-trivial examples of this type of …
Problem The first known prime found to exceed one million digits was discovered in 1999, and is a Mersenne prime of the form 26972593-1; it contains exactly 2,098,960 digits. Subsequently other Mersenne primes, of the form 2p-1, have been found which contain more digits. However, in 2004 there was found a massive non-Mersenne prime which …
Problem The first two consecutive numbers to have two distinct prime factors are: 14 = 2 x 7 15 = 3 x 5 The first three consecutive numbers to have three distinct prime factors are: 644 = 2² x 7 x 23 645 = 3 x 5 x 43 646 = 2 x 17 x …
Problem Pentagonal numbers are generated by the formula, Pn=n(3n-1)/2. The first ten pentagonal numbers are: 1, 5, 12, 22, 35, 51, 70, 92, 117, 145, … It can be seen that P4 + P7 = 22 + 70 = 92 = P8. However, their difference, 70 – 22 = 48, is not pentagonal. Find the …
Problem 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 + 2×12 15 = 7 + 2×22 21 = 3 + 2×32 25 = 7 + 2×32 27 = 19 + 2×22 33 = 31 + 2×12 …
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