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