Problem Solving

Gmat_mission wrote:With # and & each representing different digits in the problem below, the difference between #&& and ## is 667. What is the value of &?

#&&
- ##
_____
667

(A) 3
(B) 4
(C) 5
(D) 8
(E) 9

[

We see that # (in the hundreds digit) represents either 6 (if there is no borrowing from the tens digit) or 7 (if there is borrowing). However, no matter if it's 6 or 7, it's also the tens digit of the subtrahend, and since the tens digit of the difference is also 6, there must be borrowing. So # must be 7. In that case, & (in the tens digit) is either 3 or 4. It can be 3 and by borrowing from 7 (in the hundreds place) becomes 13 and 13 - 7 = 6 (if there is no borrow from the units digit). It can be 4 and by borrowing from 7 (in the hundreds place) and by lending 1 to the units digit becomes 13 and 13 - 7 = 6. We can see which one is correct by checking both cases:

733 - 77 = 656 (This is not correct.)

744 - 77 = 667 (This is correct.)

Answer: B