Difference between revisions of "1995 AIME Problems/Problem 10"

Latest revision as of 11:34, 14 June 2024

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.

$\begin{tabular}{|r||r|r|r|r|r|} \hline 1 & 43 & 85&&& \\ 2&44&&&& \\ 3&45&&&& \\ 5&47&89&131&173&215 \\ 7&49&&&& \\ 11&53&95&&& \\ 13&55&&&& \\ 17&59&101&143&& \\ 19&61&103&145&& \\ 23&65&&&& \\ 29&71&113&155&& \\ 31&73&115&&& \\ 37&79&121&&& \\ 41&83&125&&& \\ \hline \end{tabular}$

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

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

Solution 2

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 $\boxed{215}$ .

@@ Line 1: / Line 1: @@
 == Problem ==
+What is the largest positive integer that is not the sum of a positive integral multiple of <math>42</math> and a positive composite integer?
 == Solution ==
+The requested number <math>\mod {42}</math> must be a prime number. Also, every number that is a multiple of <math>42</math> 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 <math>42</math> and the numbers that are multiples of <math>42</math> greater than them, until they reach a composite number.
+<cmath>
+\begin{tabular}{|r||r|r|r|r|r|}
+\hline
+& 43 & 85&&& \\
+&44&&&& \\
+&45&&&& \\
+&47&89&131&173&215 \\
+&49&&&& \\
+&53&95&&& \\
+&55&&&& \\
+&59&101&143&& \\
+&61&103&145&& \\
+&65&&&& \\
+&71&113&155&& \\
+&73&115&&& \\
+&79&121&&& \\
+&83&125&&& \\
+\hline
+\end{tabular}</cmath>
+Since <math>\boxed{215}</math> is the greatest number in the list, 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>.
+~minor edit [[User: Yiyj1|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 <math>5 \mod 42</math>. Specifically, <math>5 * 42 + 5</math>.
+-jackshi2006
+== Solution 2 ==
+Let our answer be <math>n</math>. Write <math> n = 42a + b </math>, where <math>a, b</math> are positive integers and <math> 0 \leq b < 42 </math>. Then note that <math> b, b + 42, ... , b + 42(a-1) </math> are all primes.
+If <math>b</math> is <math>0\mod{5}</math>, then <math>b = 5</math> because <math>5</math> is the only prime divisible by <math>5</math>. We get <math> n = 215</math> as our largest possibility in this case.
+If <math>b</math> is <math>1\mod{5}</math>, then <math>b + 2 \times 42</math> is divisible by <math>5</math> and thus <math>a \leq 2</math>. Thus, <math>n \leq 3 \times 42 = 126 < 215</math>.
+If <math>b</math> is <math>2\mod{5}</math>, then <math>b + 4 \times 42</math> is divisible by <math>5</math> and thus <math>a \leq 4</math>. Thus, <math>n \leq 5 \times 42 = 210 < 215</math>.
+If <math>b</math> is <math>3\mod{5}</math>, then <math>b + 1 \times 42</math> is divisible by <math>5</math> and thus <math>a = 1</math>. Thus, <math>n \leq 2 \times 42 = 84 < 215</math>.
+If <math>b</math> is <math>4\mod{5}</math>, then <math>b + 3 \times 42</math> is divisible by <math>5</math> and thus <math>a \leq 3</math>. Thus, <math>n \leq 4 \times 42 = 168 < 215</math>.
+Our answer is <math>\boxed{215}</math>.
 == See also ==
-* [[1995 AIME Problems]]
+{{AIME box|year=1995|num-b=9|num-a=11}}
+[[Category:Intermediate Number Theory Problems]]
+{{MAA Notice}}

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