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Printimc
number theory intermediate
Problem
Steve wrote the digits , , , , and in order repeatedly from left to right, forming a list of digits, beginning He then erased every third digit from his list (that is, the rd, th, th, digits from the left), then erased every fourth digit from the resulting list (that is, the th, th, th, digits from the left in what remained), and then erased every fifth digit from what remained at that point. What is the sum of the three digits that were then in the positions ?
(A)
(B)
(C)
(D)
Solution
Note that cycles exist initially and after each round of erasing. Let the parentheses denote cycles. It follows that: 1. Initially, the list has cycles of length 2. To find one cycle after the first round of erasing, we need one cycle of length before erasing. So, we first group copies of the current cycle into one, then erase: As a quick confirmation, one cycle should have length at this point. 3. To find one cycle after the second round of erasing, we need one cycle of length before erasing. So, we first group copies of the current cycle into one, then erase: As a quick confirmation, one cycle should have length at this point. 4. To find one cycle after the third round of erasing, we need one cycle of length before erasing. We already have it here, so we erase: As a quick confirmation, one cycle should have length at this point. Since are congruent to modulo respectively, the three digits in the final positions are respectively: Therefore, the answer is
Final answer
D