From the positive integers 1-100, create a set of integers, such that each pair of integers has a different lowest common multiple. For example, {2, 3, 4, 5} would work because all six LCMs are unique, but {2, 5, 6, 9} would not because LCM(2,9) = LCM(6,9) = 18. Try to find the largest such set. You could just use {all primes}, but can you find larger?
Extension: Find a set of 10 integers, such that each pair of integers has a different LCM, and the largest element of the set is as small as possible.
Source: Original.
The largest set found was length 36 by several people. Kirk Bresniker found 147 different length-36 solutions. From the results he found,
The smallest sum is: 2081 1 15 25 29 31 32 36 37 38 41 42 46 47 48 49 52 53 54 55 59 61 67 68 71 73 77 79 81 83 85 86 88 89 91 95 97 The biggest sum is: 2574 1 28 34 43 49 53 54 55 59 61 63 64 65 67 71 73 74 75 76 78 79 81 82 83 85 87 88 89 91 92 93 94 95 96 97 99
The set of 10 integers with minimum largest value was found by many solvers:
2 3 4 5 7 9 11 13 17 19