Difference between revisions of "2006 AIME I Problems/Problem 3"
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== Solution == | == Solution == | ||
− | The number can be represented as <math>10^na+b</math>, where <math> a </math> is the leftmost digit, and <math> b </math> is the rest of the number. We know that <math>b=\frac{10^na+b}{29} \implies 28b=2^2\times7b=10^na</math>. Thus <math> a </math> has to be 7 since <math> 10^n </math> can not have 7 as a factor, and the smallest <math> 10^n </math> can be and have a factor of <math> 2^2 </math> is <math> 10^2=100. </math> We find that <math>b=25</math>, so the number is <math>725</math>. | + | The number can be represented as <math>10^na+b</math>, where <math> a </math> is the leftmost digit, and <math> b </math> is the rest of the number.* We know that <math>b=\frac{10^na+b}{29} \implies 28b=2^2\times7b=10^na</math>. Thus <math> a </math> has to be 7 since <math> 10^n </math> can not have 7 as a factor, and the smallest <math> 10^n </math> can be and have a factor of <math> 2^2 </math> is <math> 10^2=100. </math> We find that <math>b=25</math>, so the number is <math>725</math>. |
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+ | *It is quite obvious that <math>n=2</math>, since the desired number can't be single or double digit, and cannot exceed 999. From <math>100a+b=29b</math>, proceed as above. | ||
== See also == | == See also == |
Revision as of 05:43, 15 August 2011
Problem
Find the least positive integer such that when its leftmost digit is deleted, the resulting integer is of the original integer.
Solution
The number can be represented as , where is the leftmost digit, and is the rest of the number.* We know that . Thus has to be 7 since can not have 7 as a factor, and the smallest can be and have a factor of is We find that , so the number is .
- It is quite obvious that , since the desired number can't be single or double digit, and cannot exceed 999. From , proceed as above.
See also
2006 AIME I (Problems • Answer Key • Resources) | ||
Preceded by Problem 2 |
Followed by Problem 4 | |
1 • 2 • 3 • 4 • 5 • 6 • 7 • 8 • 9 • 10 • 11 • 12 • 13 • 14 • 15 | ||
All AIME Problems and Solutions |