Difference between revisions of "1995 AIME Problems/Problem 10"
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| − | <math>\boxed{215}</math> is the greatest number in the list, so it is the answer. Note that considering <math>\mod {5}</math> would have shortened the search, since <math>\text{gcd}(5,42)=1</math>, and so within <math>5</math> numbers at least one must be divisible by <math>5</math>. | + | <math>\boxed{215}</math> is the greatest number in the list, so it is the answer. Note that considering <math>\mod {5}</math> would have shortened the search, since <math>\text{gcd}(5,42)=1</math>, and so within <math>5</math> numbers at least one must be divisible by <math>5</math>. |
== See also == | == See also == | ||
Revision as of 00:43, 30 December 2015
Problem
What is the largest positive integer that is not the sum of a positive integral multiple of 
and a positive composite integer?
Solution
The requested number 
must be a prime number. Also, every number that is a multiple of greater than that prime number must also be prime, except for the requested number itself. So we make a table, listing all the primes up to and the numbers that are multiples of greater than them, until they reach a composite number.
![\[\begin{tabular}{|r||r|r|r|r|r|} \hline 2&44&&&& \\ 3&45&&&& \\ 5&47&89&131&173&215 \\ 7&49&&&& \\ 11&53&95&&& \\ 13&55&&&& \\ 17&59&101&143&& \\ 19&61&103&145&& \\ 23&65&&&& \\ 29&71&113&155&& \\ 31&73&115&&& \\ 37&79&121&&& \\ 41&83&125&&& \\ \hline \end{tabular}\]](http://latex.artofproblemsolving.com/b/3/0/b300b2f4700c487eeca79b64f8b0792ef6475fdf.png)
is the greatest number in the list, so it is the answer. Note that considering 
would have shortened the search, since 
, and so within 
numbers at least one must be divisible by .
See also
| 1995 AIME (Problems • Answer Key • Resources) | ||
| Preceded by Problem 9 |
Followed by Problem 11 | |
| 1 • 2 • 3 • 4 • 5 • 6 • 7 • 8 • 9 • 10 • 11 • 12 • 13 • 14 • 15 | ||
| All AIME Problems and Solutions | ||
The problems on this page are copyrighted by the Mathematical Association of America's American Mathematics Competitions. 
