Is it possible?
Number Theory
Level
3
In my invented game of darts, each dart that you throw will score you either 8 or 15 points. If you are allowed an unlimited number of shots, what is the largest score that
cannot
be attained?
The answer is 97.
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1 solution
Every possible positive integer i that can be formed with the numbers 8 and 15 should be in the form: i = 1 5 m + 8 n , where m and n are positive integers.
To find the largest number that cannot be formed, we can first start with a number that can be formed and search downwards.
To generalize this process, we can use induction. Suppose we have a number i that can be expressed in the form above. We want to determine if i-1 can be expressed in the form above. We have two cases to consider.
c a s e 1 : n > = 2
i − 1 = 1 5 m + 8 n − 1
= 1 5 m + 8 ( n − 2 ) + 1 6 − 1
= 1 5 ( m + 1 ) + 8 ( n − 2 )
= 1 5 p + 8 q where p = m+1, q = n-2 and p, q are positive integers
c a s e 2 : m > = 7
i − 1 = 1 5 m + 8 n − 1
= 1 5 ( m − 7 ) + 8 n + 1 0 5 − 1
| = 1 5 ( m − 7 ) + 8 ( n + 1 3 ) where p = m-7, q = n+13 and p, q are positive integers
Hence it is proven that any number can be formed if and only if both the conditions of n>=2 and m >=7, are met. Our only option is to find the largest number that does not meet both of these conditions. Thus, the answer is 1 5 ( 7 − 1 ) + 8 ( 2 − 1 ) = 9 7