dartboard math problem?

A dartboard has two regions for earning points: 5 points and 7 points. What is the largest score that is impossible to earn using this dartboard?

Comments

  • It is possible to earn every number of the form:

    5x is possible starting with 0

    5x + 7 = 5(x + 1) + 2 starting with 7

    5x + 2*7 = 5x + 14 = 5(x + 2) + 4 starting with 14

    5x + 3*7 = 5x + 21 = 5(x + 4) + 1 starting with 21

    5x + 4*7 = 5x + 28 = 5(x + 5) + 3 starting with 28

    At 28 we now have a series of numbers with every possible remainder (0 thru 4) of a multiple of 5. So every number ≥ 28 can be earned.

    Now we have to test backwards from 27 until we find a number impossible to earn.

    27 = 4*5 + 7

    26 = 5 + 3*7

    25 = 5*5

    24 = 2*5 + 2*7

    23 is not possible to earn. It is the largest number not possible to earn.

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