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

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==Solution 2 ==
 
==Solution 2 ==
  
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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 0-9, n is equal to <math>(d)+10(d+1)+100(d+2)+1000(d+3)</math> or <math>1111d +3210</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) do not work because of, n 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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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 ==

Revision as of 22:16, 21 July 2018

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

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 $(0 + 1 + 2 + 3 + 4 + 5 + 6) + 7\cdot28 = \boxed{217}$, our answer.


Solution 2

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($d$) $0-6$($7-9$ do not work because a digit cannot be greater than 9) do not work because of, n is equal to $(d)+10(d+1)+100(d+2)+1000(d+3)$ or $1111d +3210$. Now we try this number for $d=0$. When $d=0$, $n=3210$ and $3210$ when divided by $37$ has a remainder of 28. We now notice that every time you increase $d$ by $1$ you increase $n$ by $1111$ and $1111$ has remainder $1$ when divided by $37$. Thus, the remainder increases by $1$ every time you increase $d$ by $1$. Thus,

When $d=0$, the remainder equals 28 When $d=1$, the remainder equals 29 When $d=2$, the remainder equals 30 When $d=3$, the remainder equals 31 When $d=4$, the remainder equals 32 When $d=5$, the remainder equals 33 When $d=6$, the remainder equals 34

Thus the sum of the remainders is equal to $28+29+30+31+32+33+34$ which is equal to $217$.

See also

2004 AIME I (Problems • Answer Key • Resources)
Preceded by
First Question 		Followed by
Problem 2
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All AIME Problems and Solutions

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