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 $5$.

Second Solution

Let our answer be $n$. Write $n = 42a + b$, where $a, b$ are positive integers and $0 \leq b < 42$. Then note that $b, b + 42, ... , b + 42(a-1)$ are all primes.

If $b$ is $0\mod{5}$, then $b = 5$ because $5$ is the only prime divisible by $5$. We get $n = 215$ as our largest possibility in this case.

If $b$ is $1\mod{5}$, then $b + 2 \times 42$ is divisible by $5$ and thus $a \leq 2$. Thus, $n \leq 3 \times 42 = 126 < 215$.

If $b$ is $2\mod{5}$, then $b + 4 \times 42$ is divisible by $5$ and thus $a \leq 4$. Thus, $n \leq 5 \times 42 = 210 < 215$.

If $b$ is $3\mod{5}$, then $b + 1 \times 42$ is divisible by $5$ and thus $a = 1$. Thus, $n \leq 2 \times 42 = 84 < 215$.

If $b$ is $4\mod{5}$, then $b + 3 \times 42$ is divisible by $5$ and thus $a \leq 3$. Thus, $n \leq 4 \times 42 = 168 < 215$.

Our answer is $\boxed{215}$.

See also

1995 AIME (ProblemsAnswer KeyResources)
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

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