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Austrian Mathematical Olympiad

Combinatorics Difficulty 4.9 AIME Find the answer Austria

Anthony writes down in order all positive integers which are divisible by $2$ . Bertha writes down in order all positive integers which are divisible by $3$ . Claire writes down in order all positive integers which are divisible by $4$ . Orderly Dora writes all numbers written by the other three. Thereby she puts them in order by size and does not repeat a number. What is the $2017$ th number in her list?
(Richard Henner)

Your answer

A number or a short expression. Fractions can be typed as 3/2, and spacing doesn't matter.

Solution

Dora can ignore Claire's numbers, since Anthony has already written them all. Considering the numbers up to $3000$ we see that Anthony has already written down $1500$ of them and Bertha has written $1000$ of them, $500$ of which have been written twice, which are ignored by Dora in their second occurrence. Hence Dora denotes exactly $2000$ numbers up to $3000$ . The next $17$ numbers written by Dora are $3002$ , $3003$ , $3004$ , $3006$ , $3008$ , $3009$ , $3010$ , $3012$ , $3014$ , $3015$ , $3016$ , $3018$ , $3020$ , $3021$ , $3022$ , $3024$ , $3026$ . Thus $3026$ is the $2017$ th number on Dora's list.

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