1995 AIME Problems/Problem 10

Problem

What is the largest positive integer that is not the sum of a positive integral multiple of $42$ and a positive composite integer?

Solution

The requested number $\mod {42}$ must be a prime number. Also, every number that is a multiple of $42$ 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 $42$ and the numbers that are multiples of $42$ greater than them, until they reach a composite number.

$\begin{tabular}{|r||r|r|r|r|r|} \hline 2&44&&&& \\ 3&45&&&& \\ 5&47&89&131&173&215 \\ 7&49&&&& \\ 11&53&95&&& \\ 13&55&&&& \\ 17&59&101&143&& \\ 19&61&103&145&& \\ 23&65&&&& \\ 29&71&113&155&& \\ 31&73&115&&& \\ 37&79&121&&& \\ 41&83&125&&& \\ \hline \end{tabular}$

$\boxed{215}$ is the greatest number in the list, so it is the answer. Note that considering $\mod {5}$ would have shortened the search, since $\text{gcd}(5,42)=1$ , and so within $5$ numbers at least one must be divisible by $5$ .

1995 AIME (Problems • Answer Key • Resources)
Preceded by Problem 9	Followed by Problem 11
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1995 AIME Problems/Problem 10

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