Full working out - process of elimination It seems likely there are either only two possible answers or of those possible only two use all the digits between them without repetition. 0, 1, 2, 3, 4, 5, 6, 7, 8, 9 lowest two digit number is 10, highest is 98 23 & 37 e.g. are prime numbers - useful if ending in seven is required as it reduces choices all available numbers: 10, 12, 13, 14, 15, 16, 17, 18, 19, 20, 21, 23, 24, 25, 26, 27, 28, 29, 30, 31, 32, 34, 35, 36, 37, 38, 39, 40, 41, 42, 43, 45, 46, 47, 48, 49, 50, 51, 52, 53, 54, 56, 57, 58, 59 60, 61, 62, 63, 64, 65, 67, 68, 69, 70, 71, 72, 73, 74, 75, 76, 78, 79, 80, 81, 82, 83, 84, 85, 86, 87, 89, 90, 91, 92, 93, 94, 95, 96, 97, 98. One of these is the required number ending in seven 17, 27, 37, 47, 57, 67, 87, 97 & as well as those others listed it also cancels use of 7, 70, 71, 72, 73, 74, 75, 76, 78, 79. It seems likely you should start looking at the number ending in seven. So of the non repeating two-digit numbers, 17 & 34, 17 & 68 17 & 85 27 & 54, 27 & 81 57 & 19 87 & 29 only a) 34 \ 17 = 2, b) 68 \ 17 = 4 c) 81 \ 27 = 3 d) 57 \ 19 = 3 e) 87 \ 29 = 3 have non repeating answers. Then, a) 12347, leaves 56890 b) 14678 leaves 23590 c) 12378 leaves 45690 d) 13579 leaves 24680 e) 23789 leaves 14560 ...which should now already start becoming obvious for a) 56890, 98, 96, 95, 90 89, 86, 85, 80 69, 68, 65, 60 59, 58, 56, 50 for b) 23590 95, 93, 92, 90 59, 53, 52, 50 39, 35, 32, 30 29, 25, 23, 20 for c) 45690 96, 95, 94, 90 69, 65, 64, 60 59, 56, 54, 50 49, 46, 45, 40 for d) 13579 97, 95, 93, 91 79, 75, 73, 71 59, 57, 53, 51 39, 37, 35, 31 19, 17, 15, 13 for e) 23789 98, 97, 93, 92 89, 87, 83, 82 79, 78, 73, 72 39, 38, 37, 32 29, 28, 27, 23 ...Of which the easiest ones to look at end in 0 and can not be a multiple of ten (9 & 90, 2 & 20), nor can any nuber be divided by one ending in 0 as it would also have to end in zero: __ \ ?0 is only a whole number for ?0 \ ?0 which is repetition. a) 56890, 98, 96, 95, 90 89, 86, 85, 80 69, 68, 65, 60 59, 58, 56, 50 using 90 leaves only 86, 85, 68, 65, 58, 56 - none are factors. - Immediately you can say the same for 80, 60 & 50 as the two digit numbers required are 40, 30, 25, 20, 15, & 12 b) 23590 95, 93, 92, 90 59, 53, 52, 50 39, 35, 32, 30 29, 25, 23, 20 using 90 leaves 53, 53, 35, 32, 25, 23 - none are factors for 50, 30 & 20 you need 25, 15, 10, none are factors or have repetition for c) 45690 96, 95, 94, 90 69, 65, 64, 60 59, 56, 54, 50 49, 46, 45, 40 for 90, 60, 50 & 40 you need 45, 30, 25, 20 & 10 which is almost promising to yield but for the fact 90 \ 45 = 2 & 2 is not in the set Then we look for any numbers that are primes - if their multiples are available all between 0-100 2, 3, 5, 7, 11, 13, 17, 19, 23, 29, 31, 37, 41, 43, 47, 53, 59, 61, 67, 71, 73, 79, 83, 89, 97 for a) 56890, 98, 96, 95, 90 89, 86, 85, 80 69, 68, 65, 60 59, 58, 56, 50 so 89 & 59 are both factors only of at least three digit numbers for b) 23590 95, 93, 92, 90 59, 53, 52, 50 39, 35, 32, 30 29, 25, 23, 20 so 59, 53 not allowed, 29 needs 58 or 87, 23 needs 46, 69 or 92 for c) 45690 96, 95, 94, 90 69, 65, 64, 60 59, 56, 54, 50 49, 46, 45, 40 so just 59 & not allowed for d) 13579 97, 95, 93, 91 79, 75, 73, 71 59, 57, 53, 51 39, 37, 35, 31 19, 17, 15, 13 so 97, 79, 73, 71, 59, 53 all at least three digit multiples 37, 31, 19, 17 & 13 need 74, 62, 93, 38, 57*, 76, 95, 34, 51, 68, 85, 26, 39, 52, 65, 78, 91 *of which 57 \ 19 = 3 is the only answer :) won't bother with this then for e) 23789 98, 97, 93, 92 89, 87, 83, 82 79, 78, 73, 72 39, 38, 37, 32 29, 28, 27, 23