# The pages of a book are numbered consecutively: 1, 2, 3, 4 and so on. No pages are missing. If in the page numbers the digit 3 occurs exactly 99 times, what is the number of the last page?

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by evanex

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by evanex

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Page numbers with the digit 3 occurring in them are :

03, 13, 23, 30, 31,32,33, 34, 35, 36, 37, 38, 39, ,43,53, 63, 73,83,93,

1 + 03,13, 23, 30, 31, 32, 33, 34,35, 36, 37, 38, 39, 43, 53, 63, 73, 83, 93,

2 +

3 +

4 + ...

so from 00 to 99: we have 20 of digit 3.

100 to 199 20 total up to 199 : 40

200 to 299 : 20 total up to 299: 60

300 to 399 : 20 + 100 total up to 399 : 180

From 300: 309 : we have 10 so total 70 of digit 3.

310 to 319 : 10 : 80

320 to 329 : 10: 90

So now, 330, 331, 332, 333, : there are 9 of the digit 3.

So the answer is 333.

03, 13, 23, 30, 31,32,33, 34, 35, 36, 37, 38, 39, ,43,53, 63, 73,83,93,

1 + 03,13, 23, 30, 31, 32, 33, 34,35, 36, 37, 38, 39, 43, 53, 63, 73, 83, 93,

2 +

3 +

4 + ...

so from 00 to 99: we have 20 of digit 3.

100 to 199 20 total up to 199 : 40

200 to 299 : 20 total up to 299: 60

300 to 399 : 20 + 100 total up to 399 : 180

From 300: 309 : we have 10 so total 70 of digit 3.

310 to 319 : 10 : 80

320 to 329 : 10: 90

So now, 330, 331, 332, 333, : there are 9 of the digit 3.

So the answer is 333.