Question

How many positive integers less than 1000 have the property that the sum of the

digits of each such number is divisible by 7 and the number itself is divisible by 3

Answer 1

one way is

coefficient of x21 : (x0 + x1 + x2 +......... + x9)3

:

10 3

3 (1– x ) (1– x)

(1 – 3x10 + 3x20 +.....)(1 – x)–3 : 3 + 21–1C21 – 3.3+11–1C11 + 3.3 = 23C21 – 3.13C11 + 9 = 2 2223 

–

2 12133 

+ 9

= 23 × 11 – 39 × 6 + 9 = 253 – 234 + 9 = 28 Ans.

OR

Sum of digit divisible by 7  S and number itself divisible by 3  Sum of number should be 21  So numbers are (1) 399 (2) 498 (3) 489 (4) 579 (5) 597 (6) 588 (7) 669 (8) 678 (9) 687 (10) 696 (11) 759 (12) 768 (13) 777 (14) 786 (15) 795 (16) 849 (17) 858 (18) 867 (19) 876 (20) 885 (21) 894 (22) 939 (23) 948 (24) 957 (25) 966 (26) 975 (27) 984 (28) 993 So total number of integer are  28