Difference between revisions of "2018 AMC 10A Problems/Problem 17"
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If we start with <math>1</math>, we can include nothing else, so that won't work. | If we start with <math>1</math>, we can include nothing else, so that won't work. | ||
| − | If we start with <math>2</math>, we would have to include every odd number to fill out the set, but then <math>3</math> and <math>9</math> would violate the rule, so that won't work. | + | If we start with <math>2</math>, we would have to include every odd number except <math>1</math> to fill out the set, but then <math>3</math> and <math>9</math> would violate the rule, so that won't work. |
Experimentation with <math>3</math> shows it's likewise impossible. You can include <math>7</math>, <math>11</math>, and either <math>5</math> or <math>10</math> (which are always safe). But after adding either <math>4</math> or <math>8</math> we have nowhere else to go. | Experimentation with <math>3</math> shows it's likewise impossible. You can include <math>7</math>, <math>11</math>, and either <math>5</math> or <math>10</math> (which are always safe). But after adding either <math>4</math> or <math>8</math> we have nowhere else to go. | ||
Revision as of 21:06, 8 February 2018
Problem
Let 
be a set of 6 integers taken from 
with the property that if 
and 
are elements of with 
, then is not a multiple of . What is the least possible values of an element in 
Solution
If we start with 
, we can include nothing else, so that won't work.
If we start with 
, we would have to include every odd number except to fill out the set, but then 
and 
would violate the rule, so that won't work.
Experimentation with shows it's likewise impossible. You can include 
, 
, and either 
or 
(which are always safe). But after adding either 
or 
we have nowhere else to go.
Finally, starting with , we find that the sequence 
works, giving us 
.
(Random_Guy)
See Also
| 2018 AMC 10A (Problems • Answer Key • Resources) | ||
| Preceded by Problem 16 |
Followed by Problem 18 | |
| 1 • 2 • 3 • 4 • 5 • 6 • 7 • 8 • 9 • 10 • 11 • 12 • 13 • 14 • 15 • 16 • 17 • 18 • 19 • 20 • 21 • 22 • 23 • 24 • 25 | ||
| All AMC 10 Problems and Solutions | ||
| 2018 AMC 12A (Problems • Answer Key • Resources) | |
| Preceded by Problem 11 |
Followed by Problem 13 |
| 1 • 2 • 3 • 4 • 5 • 6 • 7 • 8 • 9 • 10 • 11 • 12 • 13 • 14 • 15 • 16 • 17 • 18 • 19 • 20 • 21 • 22 • 23 • 24 • 25 | |
| All AMC 12 Problems and Solutions | |
The problems on this page are copyrighted by the Mathematical Association of America's American Mathematics Competitions. 
