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Olympiad workbook
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IOQM
Chapter
Basics of Mathematics

Olympiad workbook · IOQM · Basics of Mathematics

How many positive integers less than 1000 have the property that...

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 ?

Correct answer

28

Step-by-step solution

coefficient of x21:x^{21}:\left(x^0+x^1+x^2+\ldots \ldots \ldots+x^9\right)^3 :(1−x10)3(1−x)3(1−3x10+3x20+…..)(1−x)−3:3+21−1C21−3⋅3+11−1C11+3.3=23C21−3⋅13C11+9=23×222−3×13×122+9=23×11−39×6+9=253−234+9=28 Ans. \begin{aligned} & : \frac{\left(1-x^{10}\right)^3}{(1-x)^3} \\ & \left(1-3 x^{10}+3 x^{20}+\ldots . .\right)(1-x)^{-3} \\ & :{ }^{3+21-1} C_{21}-3 \cdot{ }^{3+11-1} C_{11}+3.3 \\ & ={ }^{23} C_{21}-3 \cdot{ }^{13} C_{11}+9=\frac{23 \times 22}{2}-\frac{3 \times 13 \times 12}{2}+9 \\ & =23 \times 11-39 \times 6+9 \\ & =253-234+9=28 \text { Ans. } \end{aligned}
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