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Difference between revisions of "2004 AIME I Problems/Problem 1"

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The digits of a positive integer <math> n </math> are four consecutive integers in decreasing order when read from left to right. What is the sum of the possible remainders when <math> n </math> is divided by <math>37</math>?
 
The digits of a positive integer <math> n </math> are four consecutive integers in decreasing order when read from left to right. What is the sum of the possible remainders when <math> n </math> is divided by <math>37</math>?
  
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== Solution ==
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== Solution 1==
 
A brute-force solution to this question is fairly quick, but we'll try something slightly more clever:  our numbers have the form <math>{\underline{(n+3)}}\,{\underline{(n+2)}}\,{\underline{( n+1)}}\,{\underline {(n)}} </math><math>= 1000(n + 3) + 100(n + 2) + 10(n + 1) + n = 3210 + 1111n</math>, for <math>n \in \lbrace0, 1, 2, 3, 4, 5, 6\rbrace</math>.
 
A brute-force solution to this question is fairly quick, but we'll try something slightly more clever:  our numbers have the form <math>{\underline{(n+3)}}\,{\underline{(n+2)}}\,{\underline{( n+1)}}\,{\underline {(n)}} </math><math>= 1000(n + 3) + 100(n + 2) + 10(n + 1) + n = 3210 + 1111n</math>, for <math>n \in \lbrace0, 1, 2, 3, 4, 5, 6\rbrace</math>.
  
 
Now, note that <math>3\cdot 37 = 111</math> so <math>30 \cdot 37 = 1110</math>, and <math>90 \cdot 37 = 3330</math> so <math>87 \cdot 37 = 3219</math>. So the [[remainder]]s are all congruent to <math>n - 9 \pmod{37}</math>. However, these numbers are negative for our choices of <math>n</math>, so in fact the remainders must equal <math>n + 28</math>.
 
Now, note that <math>3\cdot 37 = 111</math> so <math>30 \cdot 37 = 1110</math>, and <math>90 \cdot 37 = 3330</math> so <math>87 \cdot 37 = 3219</math>. So the [[remainder]]s are all congruent to <math>n - 9 \pmod{37}</math>. However, these numbers are negative for our choices of <math>n</math>, so in fact the remainders must equal <math>n + 28</math>.
  
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Adding these numbers up, we get <math>(0 + 1 + 2 + 3 + 4 + 5 + 6) + 7\cdot28 = \boxed{217}</math>, our answer.
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Adding these numbers up, we get <math>(0 + 1 + 2 + 3 + 4 + 5 + 6) + 7\cdot28 = \boxed{217}</math>
  
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== Solution 2==
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There are only <math>7</math> possible values of <math>n</math>: <math>3210, 4321, 5432, \cdots, 9876</math>.
  
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==Solution 2 ==
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<math>3210</math> gives a remainder of <math>28</math> when divided by <math>37</math>. To calculate the remainders of the other integers, notice that each number is <math>1111</math> more than the previous number. Since <math>1111</math> gives a remainder of <math>1</math> when divided by <math>37</math>, the remainders of the other integers will be <math>29, 30, 31, \cdots, 34</math>.
  
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For any n we are told that it is four consecutive integers in decreasing order when read from left to right. Thus for any number(<math>d</math>) <math>0-6</math>(<math>7-9</math> do not work because a digit cannot be greater than 9), <math>n</math> is equal to <math>(d)+10(d+1)+100(d+2)+1000(d+3)</math> or <math>1111d +3210</math>. Now we try this number for <math>d=0</math>. When <math>d=0</math>, <math>n=3210</math> and <math>3210</math> when divided by <math>37</math> has a remainder of 28. We now notice that every time you increase <math>d</math> by <math>1</math> you increase <math>n</math> by <math>1111</math> and <math>1111</math> has remainder <math>1</math> when divided by <math>37</math>. Thus, the remainder increases by <math>1</math> every time you increase <math>d</math> by <math>1</math>. Thus,
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Therefore, our answer is <math>28 + 29 + 30 + 31 + 32 + 33 + 34 = \frac{28 + 34}{2} \cdot 7 = \boxed{217}</math>.
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~Viliciri
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When <math>d=0</math>, the remainder equals 28
 
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When <math>d=1</math>, the remainder equals 29
 
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When <math>d=2</math>, the remainder equals 30
 
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When <math>d=3</math>, the remainder equals 31
 
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When <math>d=4</math>, the remainder equals 32
 
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When <math>d=5</math>, the remainder equals 33
 
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When <math>d=6</math>, the remainder equals 34
 
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Thus the sum of the remainders is equal to <math>28+29+30+31+32+33+34</math> which is equal to <math>217</math>.
 
  
 
== See also ==
 
== See also ==

Latest revision as of 22:56, 2 December 2024

Problem

The digits of a positive integer $n$ are four consecutive integers in decreasing order when read from left to right. What is the sum of the possible remainders when $n$ is divided by $37$?

Solution 1

A brute-force solution to this question is fairly quick, but we'll try something slightly more clever: our numbers have the form ${\underline{(n+3)}}\,{\underline{(n+2)}}\,{\underline{( n+1)}}\,{\underline {(n)}}$$= 1000(n + 3) + 100(n + 2) + 10(n + 1) + n = 3210 + 1111n$, for $n \in \lbrace0, 1, 2, 3, 4, 5, 6\rbrace$.

Now, note that $3\cdot 37 = 111$ so $30 \cdot 37 = 1110$, and $90 \cdot 37 = 3330$ so $87 \cdot 37 = 3219$. So the remainders are all congruent to $n - 9 \pmod{37}$. However, these numbers are negative for our choices of $n$, so in fact the remainders must equal $n + 28$.

Adding these numbers up, we get $(0 + 1 + 2 + 3 + 4 + 5 + 6) + 7\cdot28 = \boxed{217}$

Solution 2

There are only $7$ possible values of $n$: $3210, 4321, 5432, \cdots, 9876$.

$3210$ gives a remainder of $28$ when divided by $37$. To calculate the remainders of the other integers, notice that each number is $1111$ more than the previous number. Since $1111$ gives a remainder of $1$ when divided by $37$, the remainders of the other integers will be $29, 30, 31, \cdots, 34$.

Therefore, our answer is $28 + 29 + 30 + 31 + 32 + 33 + 34 = \frac{28 + 34}{2} \cdot 7 = \boxed{217}$. ~Viliciri

See also

2004 AIME I (Problems • Answer Key • Resources)
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Problem 2
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