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

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Note: Please do not post problems here until after the AIME.
 
  
 
== Problem ==
 
== Problem ==
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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.
  
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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.
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-molocyxu
 
-molocyxu
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==Solution 2 (Official MAA)==
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Suppose that <math>N</math> has the required property. Then there are positive integers <math>k</math> and <math>m</math> such that <math>N = 10^4m + 2020 = k\cdot m</math>. Thus <math>(k - 10^4)m = 2020</math>, which holds exactly when <math>m</math> is a positive divisor of <math>2020.</math> The number <math>2020 = 2^2\cdot 5\cdot 101</math> has <math>12</math> divisors: <math>1, 2, 4, 5, 10, 20, 101, 202, 404, 505, 1010</math>, and <math>2020.</math> The requested sum is therefore the sum of the digits in these divisors plus <math>12</math> times the sum of the digits in <math>2020,</math> which is
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<cmath>(1+2+4+5+1+2+2+4+8+10+2+4)+12\cdot4 = 93.</cmath>
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==Solution 3==
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Note that for all <math>N \in S</math>, <math>N</math> can be written as <math>N=10000x+2020=20(500x+101)</math> for some positive integer <math>x</math>. Because <math>N</math> must be divisible by <math>x</math>, <math>\frac{20(500x+101)}{x}</math> is an integer. We now let <math>x=ab</math>, where <math>a</math> is a divisor of <math>20</math>. Then <math>\frac{20(500x+101)}{x}=(\frac{20}{a})( \frac{500x}{b}+\frac{101}{b})</math>. We know <math>\frac{20}{a}</math> and <math>\frac{500x}{b}</math> are integers, so for <math>N</math> to be an integer, <math>\frac{101}{b}</math> must be an integer. For this to happen, <math>b</math> must be a divisor of <math>101</math>. <math>101</math> is prime, so <math>b\in \left \{ 1, 101 \right \}</math>. Because <math>a</math> is a divisor of <math>20</math>, <math>a \in \left \{ 1,2,4,5,10,20\right\}</math>. So <math>x \in \left\{1,2,4,5,10,20,101,202,404,505,1010,2020\right\}</math>. Be know that all <math>N</math> end in <math>2020</math>, so the sum of the digits of each <math>N</math> is the sum of the digits of each <math>x</math> plus <math>2+0+2+0=4</math>. Hence the sum of all of the digits of the numbers in <math>S</math> is <math>12 \cdot 4 +45=\boxed{093}</math>.
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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==

Latest revision as of 13:02, 1 August 2022

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 \[(1+2+4+5+1+2+2+4+8+10+2+4)+12\cdot4 = 93.\]

Solution 3

Note that for all $N \in S$, $N$ can be written as $N=10000x+2020=20(500x+101)$ for some positive integer $x$. Because $N$ must be divisible by $x$, $\frac{20(500x+101)}{x}$ is an integer. We now let $x=ab$, where $a$ is a divisor of $20$. Then $\frac{20(500x+101)}{x}=(\frac{20}{a})( \frac{500x}{b}+\frac{101}{b})$. We know $\frac{20}{a}$ and $\frac{500x}{b}$ are integers, so for $N$ to be an integer, $\frac{101}{b}$ must be an integer. For this to happen, $b$ must be a divisor of $101$. $101$ is prime, so $b\in \left \{ 1, 101 \right \}$. Because $a$ is a divisor of $20$, $a \in \left \{ 1,2,4,5,10,20\right\}$. So $x \in \left\{1,2,4,5,10,20,101,202,404,505,1010,2020\right\}$. Be know that all $N$ end in $2020$, so the sum of the digits of each $N$ is the sum of the digits of each $x$ plus $2+0+2+0=4$. Hence the sum of all of the digits of the numbers in $S$ is $12 \cdot 4 +45=\boxed{093}$.

Video Solutions

See Also

2020 AIME I (Problems • Answer Key • Resources)
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