A certain game pays players in tokens, each of which is

[BTGmoderatorDC]

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.)

A) 2mn

B) 2mn - m - n

C) 2mn - m - n - 1

D) mn + m + n - 1

E) mn - m - n

OA B

Source: Manhattan Prep

[Scott@TargetTestPrep]

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.)

A) 2mn

B) 2mn - m - n

C) 2mn - m - n - 1

D) mn + m + n - 1

E) mn - m - n

OA B

Source: Manhattan Prep

Like the example given in the problem, we can let m = 2 and n = 3 and analyze each answer choice:

A) 2mn = 2(2)(3) = 12

However, 12 = 6m + 0n or 0m + 4n.

B) 2mn - m - n = 2(2)(3) - 2 - 3 = 7

We see that 7 = 2m + n only. We can skip choices C and E since the value of those expressions will be less than that of choice B and we are looking for the greatest possible sum.

D) mn + m + n - 1 = 2(3) + 2 + 3 - 1 = 10

However, 10 = 5m + 0n or 2m + 2n.

Answer: B