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0
70
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How many distinct positive integers can be represented as the difference of two numbers in the set {1, 3, 5, 7, 9, 11, 13, 15, 17, 19}?

 Jun 5, 2021
 #1
avatar+179 
+2

The maximum difference is 18, so the answer will naturally be between 1 and 18.  

 

It is impossible to represent an odd number difference, thus the differences must always be even.  Numbers are 2, 4, 6, 8, 10, 12, 14, 16, 18, for a total of 9 possibilities.  

 

Proof that you cannot get an odd difference with only odd numbers.  

 

If a is an odd number then it can be written as 2x+1, where x is an integer.  

 

If b is an odd number, then it can be written as 2y+1, where y is an integer.  

 

a-b (odd minus odd) = 2x-1-(2y+1) = 2x-2y.  Factoring, we have 2(x-y).  Because both x and y are integers, 2(x-y) will always be even, never odd.  

 Jun 5, 2021
 #2
avatar+121054 
+1

     ways to make a difference of

9                                                       2

8                                                       4

7                                                       6

6                                                       8

5                                                     10

4                                                     12

3                                                     14

2                                                     16

1                                                     18

 

9  distinct    positive differences

 

 

cool cool cool

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 Jun 5, 2021

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