Difference between revisions of "2004 AIME I Problems/Problem 1"
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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) | + | 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=0</math>, the remainder equals 28 |
Revision as of 22:17, 21 July 2018
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 , 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() ( 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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