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68 is a Perrin number. [1] It has an aliquot sum of 58 within an aliquot sequence of two composite numbers (68, 58,32,31,1,0) to the Prime in the 31-aliquot tree. It is the largest known number to be the sum of two primes in exactly two different ways: 68 = 7 + 61 = 31 + 37. [2]
For example, if 12 across and 7 down both have three digits and the clue for 12 across is "7 down times 2", one can work out that (i) the last digit of 12 across must be even, (ii) the first digit of 7 down must be 1, 2, 3 or 4, and (iii) the first digit of 12 across must be between 2 and 9 inclusive.
If two divisions are done, a multiple of 5 · 5=25 rather than 5 must be used, because 25 can be divided by 5 twice. So the number of coconuts that could be in the pile is k · 25 – 4. k =1 yielding 21 is the smallest positive number that can be successively divided by 5 twice with remainder 1.
Even numbers are always 0, 2, or 4 more than a multiple of 6, while odd numbers are always 1, 3, or 5 more than a multiple of 6. Well, one of those three possibilities for odd numbers causes an issue.
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The even numbers form an ideal in the ring of integers, [13] but the odd numbers do not—this is clear from the fact that the identity element for addition, zero, is an element of the even numbers only. An integer is even if it is congruent to 0 modulo this ideal, in other words if it is congruent to 0 modulo 2, and odd if it is congruent to 1 ...
By far the largest extant collection of Antique Greek riddles is Book 14 of the Greek Anthology, as preserved in Codex Parisianus suppl. Graecus 384, which contains about 50 verse riddles. [5]: 53 n. 10 [6] They are in a group of about 150 puzzles: the first fifty or so are oracles; the second fifty or so are arithmetical problems; and the ...
Another form of logic puzzle, popular among puzzle enthusiasts and available in magazines dedicated to the subject, is a format in which the set-up to a scenario is given, as well as the object (for example, determine who brought what dog to a dog show, and what breed each dog was), certain clues are given ("neither Misty nor Rex is the German Shepherd"), and then the reader fills out a matrix ...