What is the largest positive integer that is not the sum of a positive integral multiple of $42$ and a positive composite integer?
The requested number $\mod {42}$ must be a [composite](https://artofproblemsolving.com/wiki/index.php/Composite) 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.
\[\begin{tabular}{|r||r|r|r|r|r|} \hline 1 & 43 & 85&&& \\ 2&44&&&& \\ 3&45&&&& \\ 5&47&89&131&173&215 \\ 7&49&&&& \\ 11&53&95&&& \\ 13&55&&&& \\ 17&59&101&143&& \\ 19&61&103&145&& \\ 23&65&&&& \\ 29&71&113&155&& \\ 31&73&115&&& \\ 37&79&121&&& \\  41&83&125&&& \\ \hline \end{tabular}\]
Since $\boxed{215}$ is the greatest number in the list, 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 $5$.
~minor edit [Yiyj1](https://artofproblemsolving.com/wiki/index.php/User:Yiyj1)

Afterword
Basically, we are looking for a number where when a multiple of 42 is subtracted from it, the result is a prime number. Any number that ends in a 5 is not prime, except for 5 itself. Since 42 keeps the parity the same and advances odd numbers in the unit digit, then we can conclude that the sought number is $5 \mod 42$. Specifically, $5 * 42 + 5$.
-jackshi2006