1995 AIME Problems/Problem 10: Difference between revisions
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== Solution == | == 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. | |||
2:44 | |||
3:45 | |||
5:47,89,131,173,215 | |||
7:49 | |||
11:53,95 | |||
13:55 | |||
17:59,101,143,185 | |||
19:61,103,145 | |||
23:65 | |||
29:71,113,155 | |||
31:73,115 | |||
37:79,121 | |||
41:83,125 | |||
215 is the greatest number in the list, so it is the answer. | |||
== See also == | == See also == | ||
* [[1995 AIME Problems/Problem 9 | Previous problem]] | * [[1995 AIME Problems/Problem 9 | Previous problem]] | ||
* [[1995 AIME Problems/Problem 11 | Next problem]] | * [[1995 AIME Problems/Problem 11 | Next problem]] | ||
* [[1995 AIME Problems]] | * [[1995 AIME Problems]] | ||
Revision as of 17:02, 2 April 2008
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.
2:44
3:45
5:47,89,131,173,215
7:49
11:53,95
13:55
17:59,101,143,185
19:61,103,145
23:65
29:71,113,155
31:73,115
37:79,121
41:83,125
215 is the greatest number in the list, so it is the answer.
