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Difference between revisions of "2017 AIME I Problems/Problem 3"

(Created page with "We see that <math>d(n)</math> appears in cycles of <math>20</math>, adding a total of <math>70</math> each cycle. Since <math>\lfloor\frac{2017}{20}\rfloor=100</math>, we know...")
 
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We see that <math>d(n)</math> appears in cycles of <math>20</math>, adding a total of <math>70</math> each cycle.
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==Problem 3==
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Since <math>\lfloor\frac{2017}{20}\rfloor=100</math>, we know that by <math>2017</math>, there have been <math>100</math> cycles, or <math>7000</math> has been added. This can be discarded, as we're just looking for the last three digits.
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For a positive integer <math>n</math>, let <math>d_n</math> be the units digit of <math>1 + 2 + \dots + n</math>. Find the remainder when
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<cmath>\sum_{n=1}^{2017} d_n</cmath>is divided by <math>1000</math>.
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==Solution==
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We see that <math>d_n</math> appears in cycles of <math>20</math> and the cycles are <cmath>1,3,6,0,5,1,8,6,5,5,6,8,1,5,0,6,3,1,0,0,</cmath> adding a total of <math>70</math> each cycle.
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Since <math>\left\lfloor\frac{2017}{20}\right\rfloor=100</math>, we know that by <math>2017</math>, there have been <math>100</math> cycles and <math>7000</math> has been added. This can be discarded as we're just looking for the last three digits.
 
Adding up the first <math>17</math> of the cycle of <math>20</math>, we get that the answer is <math>\boxed{069}</math>.
 
Adding up the first <math>17</math> of the cycle of <math>20</math>, we get that the answer is <math>\boxed{069}</math>.
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==Video Solution==
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https://youtu.be/BiiKzctXDJg ~Shreyas S
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==See Also==
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{{AIME box|year=2017|n=I|num-b=2|num-a=4}}
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{{MAA Notice}}

Revision as of 19:00, 17 June 2020

Problem 3

For a positive integer $n$, let $d_n$ be the units digit of $1 + 2 + \dots + n$. Find the remainder when \[\sum_{n=1}^{2017} d_n\]is divided by $1000$.

Solution

We see that $d_n$ appears in cycles of $20$ and the cycles are \[1,3,6,0,5,1,8,6,5,5,6,8,1,5,0,6,3,1,0,0,\] adding a total of $70$ each cycle. Since $\left\lfloor\frac{2017}{20}\right\rfloor=100$, we know that by $2017$, there have been $100$ cycles and $7000$ has been added. This can be discarded as we're just looking for the last three digits. Adding up the first $17$ of the cycle of $20$, we get that the answer is $\boxed{069}$.

Video Solution

https://youtu.be/BiiKzctXDJg ~Shreyas S

See Also

2017 AIME I (Problems • Answer Key • Resources)
Preceded by
Problem 2 		Followed by
Problem 4
1 • 2 • 3 • 4 • 5 • 6 • 7 • 8 • 9 • 10 • 11 • 12 • 13 • 14 • 15
All AIME Problems and Solutions

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