Given th alphametic where each letter represents a different digit the value of c in the least value for ABCD?

Question

is
ABCD
-BCDA
------------------
1512
------------------

a)1
b)2
c)6
d)7

Answer 1

let 'BCD' = x
then the given expression can be written as:

1000A+x - 10x - A = 1512
(ABCD = 1000A + BCD; and BCDA = BCD0 + A)

or 999A - 9x = 1512

we look for the least value for ABCD; that is, minimum A....
A cannot be 0 or 1; the minimum is 2. so that

999*2 - 9x=1512 or x = (1998-1512)/9 = 486/9 = "054" = "BCD"

hence c = 5

so we have the relation:
2054 - 0542 = 1512

since the option '5' is not given, we assume that x should not start with '0'

we look for next solution....ie. A=3
then we have 999*3 - 1512 = 9x or x = 2997-1512/9 = 1485/9
= 165 in which case c='6'

note: a recent olympiad problem!

Answer 2

c) 6

3165 <- A can't be 1 or 2
-1653
--------
1512