SOLUTION: Melissa wrote down several distinct positive integers, none of them exceeding 100. Their product was not divisible by 18. At most how many numbers could she have written?

Algebra ->  Customizable Word Problem Solvers  -> Numbers -> SOLUTION: Melissa wrote down several distinct positive integers, none of them exceeding 100. Their product was not divisible by 18. At most how many numbers could she have written?       Log On

Ad: Over 600 Algebra Word Problems at edhelper.com


    

Question 1070519: Melissa wrote down several distinct positive integers, none of
them exceeding 100. Their product was not divisible by 18. At
most how many numbers could she have written?
Answer by Edwin McCravy(20060) About Me  (Show Source):
You can put this solution on YOUR website!
18 = 2*3*3

So to avoid getting a product divisible by 18, she could either 
avoid multiples of 2 or 3.  There are fewer multiples of 3, so she
would write down more numbers by avoiding the multiples of 3.

The multiples of 3 not exceeding 100 are 

3,6,9,...,99.

To see how many they are, divide those all by 3

1,2,3,...,33.

So, there are 33 multiples of 3. To exclude them, 100-33 = 67.  

However she can also write ONE multiple of 3 (that isn't a multiple
of 9), without getting a multiple of 18, so that's one more she could 
write, making her list contain 68 numbers.

Answer: 68.  She might as well include 3 as her ONE multiple of 3
(that isn't a multiple of 9), so her list could go: 

1*2*3*4*5*7*8*10*11*13*...95*97*98*100.

Edwin