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Difference between revisions of "2020 AIME I Problems/Problem 4"

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Let <math>S</math> be the set of positive integers <math>N</math> with the property that the last four digits of <math>N</math> are <math>2020,</math> and when the last four digits are removed, the result is a divisor of <math>N.</math> For example, <math>42,020</math> is in <math>S</math> because <math>4</math> is a divisor of <math>42,020.</math> Find the sum of all the digits of all the numbers in <math>S.</math> For example, the number <math>42,020</math> contributes <math>4+2+0+2+0=8</math> to this total.
 
Let <math>S</math> be the set of positive integers <math>N</math> with the property that the last four digits of <math>N</math> are <math>2020,</math> and when the last four digits are removed, the result is a divisor of <math>N.</math> For example, <math>42,020</math> is in <math>S</math> because <math>4</math> is a divisor of <math>42,020.</math> Find the sum of all the digits of all the numbers in <math>S.</math> For example, the number <math>42,020</math> contributes <math>4+2+0+2+0=8</math> to this total.
  
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== Solution ==
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== Solution 1 ==
  
 
We note that any number in <math>S</math> can be expressed as <math>a(10,000) + 2,020</math> for some integer <math>a</math>. The problem requires that <math>a</math> divides this number, and since we know <math>a</math> divides <math>a(10,000)</math>, we need that <math>a</math> divides 2020. Each number contributes the sum of the digits of <math>a</math>, as well as <math>2 + 0 + 2 +0 = 4</math>. Since <math>2020</math> can be prime factorized as <math>2^2 \cdot 5 \cdot 101</math>, it has <math>(2+1)(1+1)(1+1) = 12</math> factors. So if we sum all the digits of all possible <math>a</math> values, and add <math>4 \cdot 12 = 48</math>, we obtain the answer.
 
We note that any number in <math>S</math> can be expressed as <math>a(10,000) + 2,020</math> for some integer <math>a</math>. The problem requires that <math>a</math> divides this number, and since we know <math>a</math> divides <math>a(10,000)</math>, we need that <math>a</math> divides 2020. Each number contributes the sum of the digits of <math>a</math>, as well as <math>2 + 0 + 2 +0 = 4</math>. Since <math>2020</math> can be prime factorized as <math>2^2 \cdot 5 \cdot 101</math>, it has <math>(2+1)(1+1)(1+1) = 12</math> factors. So if we sum all the digits of all possible <math>a</math> values, and add <math>4 \cdot 12 = 48</math>, we obtain the answer.
  
 
Now we list out all factors of <math>2,020</math>, or all possible values of <math>a</math>. <math>1,2,4,5,10,20,101,202,404,505,1010,2020</math>. If we add up these digits, we get <math>45</math>, for a final answer of <math>45+48=\boxed{093}</math>.
 
Now we list out all factors of <math>2,020</math>, or all possible values of <math>a</math>. <math>1,2,4,5,10,20,101,202,404,505,1010,2020</math>. If we add up these digits, we get <math>45</math>, for a final answer of <math>45+48=\boxed{093}</math>.
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-molocyxu
 
-molocyxu
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==Video solution==
 
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https://youtu.be/5b9Nw4bQt_k
 
  
 
==Solution 2 (Official MAA)==
 
==Solution 2 (Official MAA)==
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<cmath>(1+2+4+5+1+2+2+4+8+10+2+4)+12\cdot4 = 93.</cmath>
 
<cmath>(1+2+4+5+1+2+2+4+8+10+2+4)+12\cdot4 = 93.</cmath>
  
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Video solution:
 
  
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https://youtu.be/djWzRC-jGYw
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==Video Solutions==
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*https://youtu.be/5b9Nw4bQt_k
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*https://youtu.be/djWzRC-jGYw
  
 
==See Also==
 
==See Also==

Revision as of 16:46, 27 January 2021

Problem

Let $S$ be the set of positive integers $N$ with the property that the last four digits of $N$ are $2020,$ and when the last four digits are removed, the result is a divisor of $N.$ For example, $42,020$ is in $S$ because $4$ is a divisor of $42,020.$ Find the sum of all the digits of all the numbers in $S.$ For example, the number $42,020$ contributes $4+2+0+2+0=8$ to this total.

Solution 1

We note that any number in $S$ can be expressed as $a(10,000) + 2,020$ for some integer $a$. The problem requires that $a$ divides this number, and since we know $a$ divides $a(10,000)$, we need that $a$ divides 2020. Each number contributes the sum of the digits of $a$, as well as $2 + 0 + 2 +0 = 4$. Since $2020$ can be prime factorized as $2^2 \cdot 5 \cdot 101$, it has $(2+1)(1+1)(1+1) = 12$ factors. So if we sum all the digits of all possible $a$ values, and add $4 \cdot 12 = 48$, we obtain the answer.

Now we list out all factors of $2,020$, or all possible values of $a$. $1,2,4,5,10,20,101,202,404,505,1010,2020$. If we add up these digits, we get $45$, for a final answer of $45+48=\boxed{093}$.


-molocyxu

Solution 2 (Official MAA)

Suppose that $N$ has the required property. Then there are positive integers $k$ and $m$ such that $N = 10^4m + 2020 = k\cdot m$. Thus $(k - 10^4)m = 2020$, which holds exactly when $m$ is a positive divisor of $2020.$ The number $2020 = 2^2\cdot 5\cdot 101$ has $12$ divisors: $1, 2, 4, 5, 10, 20, 101, 202, 404, 505, 1010$, and $2020.$ The requested sum is therefore the sum of the digits in these divisors plus $12$ times the sum of the digits in $2020,$ which is \[(1+2+4+5+1+2+2+4+8+10+2+4)+12\cdot4 = 93.\]


Video Solutions

See Also

2020 AIME I (Problems • Answer Key • Resources)
Preceded by
Problem 3 		Followed by
Problem 5
1 • 2 • 3 • 4 • 5 • 6 • 7 • 8 • 9 • 10 • 11 • 12 • 13 • 14 • 15
All AIME Problems and Solutions

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