+1206 How many distinct positive integers can be represented as the difference of two numbers in the set \(\{1, 3, 5, 7, 9, 11, 13\}\)?
+997 The explanation is sort of complicated, and I'll try to figure out a way to simplify what I want to say into a few short words, but in the meantime, I am going to tell you that the answer I got for this problem was \(\boxed 6\).
+128448 Knockout is correct....!!!!
Note that all the possible positive differences are generated by
3 -1 = 2
5 -1 = 4
7-1 = 6
9-1 = 8
11 - 1 = 10
13 - 1 = 12
All the other positive differences are just subsets of these 6
+997 Basically, in a few short words, the explanation is:
Since the numbers in this set are part of an arithmetic sequence, every number that comes after the first term is increasing at a constant value. In our case, the numbers are increasing by 2.
Since the numbers are are in this arithmetic sequence, the difference between any two consecutive numbers is going to be 2.
Therefore, \(\boxed 2\) is our first integer.
We can get the next integer using a similar method. Since this is an arithmetic sequence as stated before, every other number that comes after the first term also increases at a constant value.
To make this easier to understand, lets take 1 and 5, 3 and 7, 5 and 9, 7 and 11, as well as 9 and 13. The positive difference between all of these pairs is 4.
Therefore, the difference between -any two integers that are split apart by one number in this sequence- , is 4.
So \(\boxed 4\) is our second integer.
Now if we were to continue on in the same fashion, we would get the positive integers of 2, 4, 6, 8, 10, and 12. Therefore, we would have \(\boxed {\boxed {6}}\) integers.
Now I know my explanation is confusing, I do have some flaws that may not be too easy to understand. If anyone could explain it better, that would be awesome!
