2018 AMC 10 A
Complete problem set with solutions and individual problem pages
Let be a set of integers taken from with the property that if and are elements of with , then is not a multiple of . What is the least possible value of an element in ? (2018 AMC 10A Problem, Question#17)
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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 no more places to go.
Finally, starting with , we find that the sequence , , , , , works, giving us .
We know that all the odd numbers (except ) can be used.
, , , , .
Now we have to choose from for the last number (out of , , , , , , ). We can eliminate , , , and , and we have , , to choose from. But wait, is a multiple of Now we have to take out either or from the list. If we take out , none of the numbers would work, but if we take out , we get: , , , , , .
So the least number is , so the answer is .
We can get the multiples for the numbers in the original set with multiples in the same original set.
: all numbers within range.
: , , , , .
: , , .
: , .
: .
: .
It will be safe to start with or since they have the smallest number of multiples as listed above, but since the question asks for the least, it will be better to try others.
Trying , we can get , , , , , . So works. Trying won't work, so the least is . This means the answer is .
