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# Help math

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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
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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
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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   .

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Jun 5, 2021