If 5a + 7b = l, where a and b are positive integers, what is the largest possible value of l for which exactly one pair of integers (a, b) makes the equation true? Choose the correct choice from the following
A) 35 B) 48 C) 69 D) 70 E) 74
5*a1 + 7*b1 = l   5*a2 + 7*b2 = l   5*(a1 - a2) = 7*(b2 - b1)   since we are dealing with integers we can assume that a1 - a2 = 7*q and b2 - b1 = 5*q where q is integer, so whenever we get a pair for (a;b) we can find another one by simply adding 7 to a and subtracting 5 from b or vice versa, subtracting 7 from a and adding 5 to b.   Lets check how it works for our numbers, starting from the largest:   E)74 = 5*12 + 7*2 (a1 = 12, b1 = 2), subtract 7 fromaand add 5 tobrespectively, so a2 = 5 and b2 = 7, second pair - bad   70)70 = 5*7 + 7*5 (a1 = 7, b1 = 5), if we add 7 toawe will have to subtract 5 from b but b can't be 0, so - no pair, if we subtract 7 froma, we'll get a = 0 which also isn't allowed - no pair, thus this is the only pair for (a;b) that works,good!, thus   70 is the answer
<result>D</result>
If 5a + 7b = l, where a and b are positive integers, what is the largest possible value of l for which exactly one pair of integers (a, b) makes the equation true?