How many of the 1000 smallest positive integers are congruent to 1 modulo 9?

You should have ALL the multiples of 9 + 1, up to 1,000. So, 999 / 9 = 111. You would use: 9n + 1, from n=1 to 111. And the sequence of these numbers would be:

 

{10, 19, 28, 37, 46, 55, 64, 73, 82, 91, 100, 109, 118, 127, 136, 145, 154, 163, 172, 181, 190, 199, 208, 217, 226, 235, 244, 253, 262, 271, 280, 289, 298, 307, 316, 325, 334, 343, 352, 361, 370, 379, 388, 397, 406, 415, 424, 433, 442, 451, 460, 469, 478, 487, 496, 505, 514, 523, 532, 541, 550, 559, 568, 577, 586, 595, 604, 613, 622, 631, 640, 649, 658, 667, 676, 685, 694, 703, 712, 721, 730, 739, 748, 757, 766, 775, 784, 793, 802, 811, 820, 829, 838, 847, 856, 865, 874, 883, 892, 901, 910, 919, 928, 937, 946, 955, 964, 973, 982, 991, 1000}. So that: (9n + 1) mod 9 = 1.