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 ?

Step-by-step solution

coefficient of x²¹: (x^0+x^1+x^2+ +x^9 )^3 aligned & : (1-x¹⁰ )^3 (1-x)^3 & (1-3 x¹⁰+3 x²⁰+ . . )(1-x)⁻³ & : 3+21-1 21 -3 3+11-1 11 +3.3 & = 23 21 -3 13 11 +9= 23 22 2 - 3 13 12 2 +9 & =23 11-39 6+9 & =253-234+9=28 Ans. aligned