Question 401349: A dart board contains a region worth 9 points and a region worth 4 points . If you are allowed to throw as many darts as you wish, then what is the largest possible total score you CANNOT get?
Answer by richard1234(7193)   (Show Source):
You can put this solution on YOUR website! We wish to find some number k that cannot be expressed in the form 9a + 4b, where a and b are positive integers.
It is apparent that if 9a + 4b is possible, then 9(a-3)+4(b+7) is possible (this is also equal to 9a + 4b + 1), as long as  . Hence, when a = 4 and b is nonnegative, then any number incremented by one is possible. This means that all numbers greater than or equal to 36 can be attained.
We list the numbers less than 36 based on the value of b in the expression 9a + 4b:
b = 0: 0, 9, 18, 27
b = 1: 4, 13, 22, 31
b = 2: 8, 17, 26, 35
b = 3: 12, 21, 30
b = 4: 16, 25, 34
b = 5: 20, 29
b = 6: 24, 33
b = 7: 28
b = 8: 32
The largest number not on this list is 23.
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