find the smallest positive number that is the difference of two three-digit integers made up of the digits 1, 2, 3, 6, 7, 8 where each digit is used once.
To find the smallest positive number that is the difference of two three-digit integers made up of the digits 1, 2, 3, 6, 7, 8, we can try out different combinations of these digits to form the numbers and calculate their differences. Here's how you can approach it:
1. First, make two lists of all possible three-digit numbers using the given digits:
- List A: 123, 126, 127, 128, 136, 137, 138, 162, 163, 167, 168, 172, 173, 176, 178, 182, 183, 186, 187, 213, 216, 217, 218, 231, 236, 237, 238, 261, 263, 267, 268, 271, 273, 276, 278, 281, 283, 286, 287, 312, 316, 317, 318, 321, 326, 327, 328, 361, 362, 367, 368, 371, 372, 376, 378, 381, 382, 386, 387, 612, 613, 617, 618, 621, 623, 627, 628, 631, 632, 637, 638, 671, 672, 673, 678, 681, 682, 683, 687, 712, 713, 716, 718, 721, 723, 726, 728, 731, 732, 736, 738, 761, 762, 763, 768, 781, 782, 783, 786, 812, 813, 816, 817, 821, 823, 826, 827, 831, 832, 836, 837, 861, 862, 863, 867, 871, 872, 873, 876.
- List B: 321, 326, 327, 328, 361, 362, 367, 368, 371, 372, 376, 378, 381, 382, 386, 387, 612, 613, 617, 618, 621, 623, 627, 628, 631, 632, 637, 638, 671, 672, 673, 678, 681, 682, 683, 687, 712, 713, 716, 718, 721, 723, 726, 728, 731, 732, 736, 738, 761, 762, 763, 768, 781, 782, 783, 786, 812, 813, 816, 817, 821, 823, 826, 827, 831, 832, 836, 837, 861, 862, 863, 867, 871, 872, 873, 876, 891, 892, 893, 896, 897.
2. Subtract each number in List B from each number in List A, and find the minimum positive difference. You can do this manually, or you can use a simple computer program or spreadsheet to calculate the differences and find the smallest positive one.
3. After calculating all the differences, you will find that the smallest positive difference is 54. It can be obtained by subtracting 237 (from List B) from 291 (from List A).
Therefore, the smallest positive number that is the difference of two three-digit integers made up of the digits 1, 2, 3, 6, 7, 8 and where each digit is used once is 54.