Problem 19 · 2019 Math Kangaroo
Stretch
Number Theory
place-valuecareful-counting
The pages of a book are numbered with 1, 2, 3, 4, 5 and so on. The digit 5 appears exactly 16 times. What is the maximum number of pages the book can have?
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Answer: B — 64
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Hint 1 of 2
Count where the digit 5 shows up as you list page numbers 1, 2, 3, ...
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Hint 2 of 2
The block 50-59 alone contributes ten 5s (the tens digit); add those to the single 5s like 5, 15, 25, ...
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Approach: count occurrences of the digit 5
- Up to 49 the 5s appear at 5, 15, 25, 35, 45: five of them.
- The block 50-59 has a 5 in every tens digit: ten more, plus the extra units-5 in 55, reaching the 16th by page 59.
- Pages 60-64 add no new 5s, but page 65 would add a 17th.
- So the most pages with exactly sixteen 5s is 64 (B).
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