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Question Number 118019 by prakash jain last updated on 14/Oct/20

$$\mathrm{How}\mathrm{many}n\mathrm{digit}\mathrm{integers}\mathrm{are}$$$$\mathrm{divisible}\mathrm{by}9?$$

Commented by prakash jain last updated on 14/Oct/20

$$d_1{d}_2....d_n\mathrm{is}n\mathrm{digit}\mathrm{number}.\mathrm{where}$$$$d_1,d_2...,d_n\mathrm{are}\mathrm{digits}.$$$$\mathrm{Express}\mathrm{result}\mathrm{in}\mathrm{terms}\mathrm{of}n.$$$$\mathrm{example}\mathrm{for}n=2$$$$18,27,36,45,54,63,72,81,90,99$$

Commented by mr W last updated on 14/Oct/20

$$9hasonlyonedigit.99isvalid.$$

Commented by prakash jain last updated on 14/Oct/20

$$\mathrm{Yes}.\mathrm{my}\mathrm{mistake}.$$

Answered by 1549442205PVT last updated on 14/Oct/20

$$\mathrm{It}\mathrm{is}\mathrm{infinite}\mathrm{many}\mathrm{because}\mathrm{the}\mathrm{condition}$$$$\mathrm{for}\mathrm{a}\mathrm{number}\mathrm{divisible}\mathrm{by}9\mathrm{that}\mathrm{is}$$$$\mathrm{the}\mathrm{sum}\mathrm{of}\mathrm{its}\mathrm{digits}\mathrm{divisible}\mathrm{by}9$$$$\mathrm{Example}:9,18,108,1008,10008,...$$

Commented by prakash jain last updated on 14/Oct/20

$$\mathrm{Sorry}\mathrm{about}\mathrm{not}\mathrm{being}\mathrm{clear}\mathrm{on}\mathrm{question}.$$$$\mathrm{Added}\mathrm{clarification}\mathrm{on}\mathrm{question}$$

Answered by mr W last updated on 14/Oct/20

$${it}^{\prime }seasierthanonemaythink.$$$$thefirstoneis{10}^{n-1}+8$$$$thesecondoneis{10}^{n-1}+17$$$$...$$$$thelastoneis{10}^n-1$$$$$$$$totally$$$$\frac{({10}^n-1)-({10}^{n-1}-1)}{9}={10}^{n-1}numbers$$

Commented by prakash jain last updated on 14/Oct/20

$$\mathrm{Thanks}.$$

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