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Question 2 : Let N, x and y be positive integers such that N = x + y, 2 < x < 10 and 14 < y < 23. If N > 25, then how many distinct values are possible for N?

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2 < x < 10

x can take any of the values from the set {3, 4, 5, 6, 7, 8, 9}

14 < y < 23

y can take any of the values from the set {15, 16, 17, 18, 19, 20, 21, 22}

The highest value N (i.e x+y) can take = 9+22 = 31. (at x = 9; y = 22)

30 can be obtained at x = 9; y = 21

29 can be obtained at x = 9; y = 20

28 can be obtained at x = 9; y = 19

27 can be obtained at x = 9; y = 18

26 can be obtained at x = 9; y = 17

25 can be obtained at x = 9; y = 16

But, x+y=25 is not the desired sum, hence the different values of x+y are {31,30,29,28,27,26}.

Hence, x+y, and thereby N can take 6 distinct values.

The question is **"Let N, x and y be positive integers such that N = x + y, 2 < x < 10 and 14 < y < 23. If N > 25, then how many distinct values are possible for N?" **

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