1990 AHSME Problems/Problem 29
Contents
Problem
A subset of the integers 
has the property that none of its members is 3 times another. What is the largest number of members such a subset can have?
Solution 1
Notice that inclusion of the integers between 
to 
inclusive is allowed as long as no integer between 
and 
inclusive is within the set. This provides a total of 
= 67$solutions.
Further analyzing the remaining integers between$ (Error compiling LaTeX. Unknown error_msg)1
10
3
3
91
9$solutions.
Thus,$ (Error compiling LaTeX. Unknown error_msg)67 + 9 = 76
\fbox{D}$
Solution 2
Write down in a column the elements 
which are indivisible by three, and then follow each one by 
![\[\begin{array}{ccccc}1&3&9&27&81\\2&6&18&54\\4&12&36\\5&15&45\\7&21&63\\8&24&72\\10&30&90\\11&33&99\\13&39\\\vdots&\vdots\end{array}\]](http://latex.artofproblemsolving.com/1/2/6/1261b43b33557578b2ffd319b00ef4d01d697fb4.png)
We can take at most 
elements from the first row, and at most 
elements from each of the next seven rows. After that we can take only 
from any following row. Thus the answer is 
the number of integers between 
and inclusive which are indivisible by three.
There are 
multiples of three in that range, so there are 
non-multiples, and 
, which is 
See also
| 1990 AHSME (Problems • Answer Key • Resources) | ||
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