A certain game pays players in tokens, each of which is worth either m points or n points, where m and n are different positive integers whose greatest common factor is 1. In terms of m and n, what is the greatest possible sum, in points, that can be paid out with only one unique combination of these tokens? (For example, if m = 2 and n = 3, then a sum of 5 points can be created using only one combination, m + n, which is a unique combination. By contrast, a sum of 11 points can be created by 4m + n or by m + 3n. This solution does not represent a unique combination; two combinations are possible.)
Choose the correct choice from the following:
A) 2mn
B) 2mn – m – n
C) 2mn – m – n – 1
D) mn + m + n – 1
E) mn – m – n
Multiple combinations are created when you can “exchange” a set of m tokens worth n points each for a set of n tokens worth m points each. In the example given in the problem, the first solution uses 4 m’s (worth 8 points total) and one n (worth 3 points) to reach 11 points. The second solution is created when 3 m’s (worth 2 points each, for a total of 6 points) are exchanged for 2 n’s (worth 3 points each, for a total of 6 points), leading to the new solution m + 3n.   The question asks for a sum with a unique solution, or a solution for which such an “exchange” is not possible. What must be true of this kind of solution?   The payout may include no more than (n – 1) m-point tokens. (If there were n of them, they could be exchanged to make a different combination.)   The payout may include no more than (m – 1) n-point tokens. (If there were m of them, they could be exchanged to make a different combination.)   The question asks for the greatest possible such sum, so use the maximum possible number of m and n tokens, as determined above:   (n – 1)(m) + (m – 1)(n)   = mn – m + mn – n   = 2mn – m – n.   Alternatively, try plugging in numbers. For instance, if m = 2 and n = 3, then you’re looking for the greatest possible sum that can be paid with a unique combination of 2- and 3-point tokens. That sum must be one of the answer choices, so check the sums in the answer choices and stop at the highest one with a unique payout combination.   With m = 2 and n = 3, the answer choices are (A) 12 points, (B) 7 points, (C) 6 points, (D) 10 points, (E) 1 point. Try them in decreasing order (because the problem asks for the greatest sum):   (A) A sum of 12 points can be paid out in three ways: six 2-point tokens; three 2-point tokens and two 3-point tokens; and four 3-point tokens. This answer is not correct.   (D) A sum of 10 points can be paid out in two ways: five 2-point tokens, or two 2-point tokens and two 3-point tokens. This answer is not correct.   (B) A sum of 7 points can be paid out only as two 2-point tokens and one 3-point token. This answer is the largest one left with a unique combination, so it is the correct answer.   The correct answer is (2mn – m – n)
<result>B</result>
A certain game pays players in tokens, each of which is worth either m points or n points, where m and n are different positive integers whose greatest common factor is 1. In terms of m and n, what is the greatest possible sum, in points, that can be paid out with only one unique combination of these tokens? (For example, if m = 2 and n = 3, then a sum of 5 points can be created using only one combination, m + n, which is a unique combination. By contrast, a sum of 11 points can be created by 4m + n or by m + 3n. This solution does not represent a unique combination; two combinations are possible.)