Difference between revisions of "2020 AMC 10B Problems/Problem 15"
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==Solution 1== | ==Solution 1== | ||
After erasing every third digit, the list becomes <math>1245235134\ldots</math> repeated. After erasing every fourth digit from this list, the list becomes <math>124235341452513\ldots</math> repeated. Finally, after erasing every fifth digit from this list, the list becomes <math>124253415251\ldots</math> repeated. Since this list repeats every <math>12</math> digits and since <math>2019,2020,2021</math> are <math>3,4,5</math> respectively in <math>\pmod{12},</math> we have that the <math>2019</math>th, <math>2020</math>th, and <math>2021</math>st digits are the <math>3</math>rd, <math>4</math>th, and <math>5</math>th digits respectively. It follows that the answer is <math>4+2+5= \boxed {\textbf{(D)} \text{ 11}}.</math> | After erasing every third digit, the list becomes <math>1245235134\ldots</math> repeated. After erasing every fourth digit from this list, the list becomes <math>124235341452513\ldots</math> repeated. Finally, after erasing every fifth digit from this list, the list becomes <math>124253415251\ldots</math> repeated. Since this list repeats every <math>12</math> digits and since <math>2019,2020,2021</math> are <math>3,4,5</math> respectively in <math>\pmod{12},</math> we have that the <math>2019</math>th, <math>2020</math>th, and <math>2021</math>st digits are the <math>3</math>rd, <math>4</math>th, and <math>5</math>th digits respectively. It follows that the answer is <math>4+2+5= \boxed {\textbf{(D)} \text{ 11}}.</math> | ||
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==Video Solution== | ==Video Solution== | ||
Revision as of 12:15, 20 June 2020
Contents
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 
?
Solution 1
After erasing every third digit, the list becomes 
repeated. After erasing every fourth digit from this list, the list becomes 
repeated. Finally, after erasing every fifth digit from this list, the list becomes 
repeated. Since this list repeats every digits and since 
are 
respectively in 
we have that the 
th, 
th, and 
st digits are the rd, th, and th digits respectively. It follows that the answer is 
Video Solution
~IceMatrix
See Also
| 2020 AMC 10B (Problems • Answer Key • Resources) | ||
| Preceded by Problem 14 |
Followed by Problem 16 | |
| 1 • 2 • 3 • 4 • 5 • 6 • 7 • 8 • 9 • 10 • 11 • 12 • 13 • 14 • 15 • 16 • 17 • 18 • 19 • 20 • 21 • 22 • 23 • 24 • 25 | ||
| All AMC 10 Problems and Solutions | ||
The problems on this page are copyrighted by the Mathematical Association of America's American Mathematics Competitions. 
