1995 AIME Problems/Problem 10
Contents
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.
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 .
Second Solution
Let our answer be . Write , where are positive integers and . Then note that are all primes.
If is , then because is the only prime divisible by . We get as our largest possibility in this case.
If is , then is divisible by and thus . Thus, .
If is , then is divisible by and thus . Thus, .
If is , then is divisible by and thus . Thus, .
If is , then is divisible by and thus . Thus, .
Our answer is .
See also
1995 AIME (Problems • Answer Key • Resources) | ||
Preceded by Problem 9 |
Followed by Problem 11 | |
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