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 …
Problem Triangle, pentagonal, and hexagonal numbers are generated by the following formulae: It can be verified that T285 = P165 = H143 = 40755. Find the next triangle number that is also pentagonal and hexagonal. Solution
Problem A googol (10100) is a massive number: one followed by one-hundred zeros; 100100 is almost unimaginably large: one followed by two-hundred zeros. Despite their size, the sum of the digits in each number is only 1. Considering natural numbers of the form, ab, where a, b < 100, what is the maximum digital sum? …
Problem If we take 47, reverse and add, 47 + 74 = 121, which is palindromic. Not all numbers produce palindromes so quickly. For example, 349 + 943 = 1292, 1292 + 2921 = 4213 4213 + 3124 = 7337 That is, 349 took three iterations to arrive at a palindrome. Although no one has …
Problem There are exactly ten ways of selecting three from five, 12345: 123, 124, 125, 134, 135, 145, 234, 235, 245, and 345 In combinatorics, we use the notation, 5C3 = 10. In general, It is not until n = 23, that a value exceeds one-million: 23C10 = 1144066. How many, not necessarily distinct, values …
Problem Euler published the remarkable quadratic formula: n² + n + 41 It turns out that the formula will produce 40 primes for the consecutive values n = 0 to 39. However, when n = 40, 402 + 40 + 41 = 40(40 + 1) + 41 is divisible by 41, and certainly when n …
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