List L: ABC, BCA, CAB
In list L above, there are 5 positive integers, where each of A, B, and C is a different nonzero digit. Which of the following is the sum of all the positive integers that MUST be factors of the sum of the integers in list L? Choose the correct choice from the following answers.
A) 47	B) 114	C) 152	D) 161	E) 488
Let's try to sum the 3 given numbers taking their place values into account:   (100A + 10B + C) + (100B + 10C + A) + (100C + 10A + B)   = 100(A + B + C) + 10(A + B + C) + (A + B + C)   = 111*(A + B + C)   We know 111 = 37 * 3 so the sum will certainly have 1, 3, 37 and 111 as factors. 1 + 3 + 37 + 111 = 152   Note: How do we know that (A + B + C) will not give us a factor that we get every time? Try and take a few cases where A, B and C have different characteristics e.g. case 1: they are all odd, case 2: they are all even with no multiple of 3 etc. We want to see if there are cases where (A+B+C) has no common factors with other cases.   Let's say A, B and C are all odd. 1+3+5 = 9. Factors 3 and 3   A, B and C are all even. 2+4+8 = 14. Factors 2 and 7   Other factors will depend on values of A, B and C. Hence there is no other factor which we MUST get.   161
<result>D</result>
List L: ABC, BCA, CAB
In list L above, there are 5 positive integers, where each of A, B, and C is a different nonzero digit. Which of the following is the sum of all the positive integers that MUST be factors of the sum of the integers in list L?