Difference between revisions of "2004 AIME I Problems/Problem 1"
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== Problem == | == Problem == | ||
− | 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 37? | + | 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>? |
== Solution == | == Solution == | ||
− | {{ | + | 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>. | ||
+ | |||
+ | Adding these numbers up, we get <math>(0 + 1 + 2 + 3 + 4 + 5 + 6) + 7\cdot28 = \boxed{217}</math> | ||
+ | |||
+ | ==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(<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, | ||
+ | |||
+ | When <math>d=0</math>, the remainder equals 28 | ||
+ | |||
+ | When <math>d=1</math>, the remainder equals 29 | ||
+ | |||
+ | When <math>d=2</math>, the remainder equals 30 | ||
+ | |||
+ | When <math>d=3</math>, the remainder equals 31 | ||
+ | |||
+ | When <math>d=4</math>, the remainder equals 32 | ||
+ | |||
+ | When <math>d=5</math>, the remainder equals 33 | ||
+ | |||
+ | When <math>d=6</math>, the remainder equals 34 | ||
+ | |||
+ | |||
+ | 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 == | ||
− | + | {{AIME box|year=2004|n=I|before=First Question|num-a=2}} | |
− | + | [[Category:Intermediate Number Theory Problems]] | |
+ | {{MAA Notice}} |
Revision as of 14:53, 27 December 2019
Contents
Problem
The digits of a positive integer are four consecutive integers in decreasing order when read from left to right. What is the sum of the possible remainders when is divided by ?
Solution
A brute-force solution to this question is fairly quick, but we'll try something slightly more clever: our numbers have the form , for .
Now, note that so , and so . So the remainders are all congruent to . However, these numbers are negative for our choices of , so in fact the remainders must equal .
Adding these numbers up, we get
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() ( do not work because a digit cannot be greater than 9), is equal to or . Now we try this number for . When , and when divided by has a remainder of 28. We now notice that every time you increase by you increase by and has remainder when divided by . Thus, the remainder increases by every time you increase by . Thus,
When , the remainder equals 28
When , the remainder equals 29
When , the remainder equals 30
When , the remainder equals 31
When , the remainder equals 32
When , the remainder equals 33
When , the remainder equals 34
Thus the sum of the remainders is equal to which is equal to .
See also
2004 AIME I (Problems • Answer Key • Resources) | ||
Preceded by First Question |
Followed by Problem 2 | |
1 • 2 • 3 • 4 • 5 • 6 • 7 • 8 • 9 • 10 • 11 • 12 • 13 • 14 • 15 | ||
All AIME Problems and Solutions |
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