SOLUTION: Pick as many numbers as you like from the first 1989 counting numbers (1, 2, 3, 1989), but don’t pick any more than once. If no two numbers can be exactly 4 apart and no two number

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Question 923948: Pick as many numbers as you like from the first 1989 counting numbers (1, 2, 3, 1989), but don’t pick any more than once. If no two numbers can be exactly 4 apart and no two numbers can be exactly 7 apart, what is the maximum amount of numbers that you can have?
Answer by Edwin McCravy(20056) About Me  (Show Source):
You can put this solution on YOUR website!
1989 is a multiple of 3.  If you pick all the multiples of 3

3,6,9,12,...,1989 

any two of them are a multiple of 3 apart and if two integers are
a multiple of 3 apart, they can neither be 4 apart nor 7 apart. 

How many of them are there?

To find out, divide them all by 3 and you have the counting
numbers from 1 through 663

1,2,3,4,...,663

so there are 663 of them.

You can also choose

1,4,7,10,...,1987 or

2,5,8,11,...,1988

There are also 663 of each of them as well.

Answer: 663

Edwin