Question

What is the digit sum of the ID number of my ATM?

• A. 9
• B. 8
• C. 1
• D. Cannot be determined

Answer 1

The correct option is A 9

The alphabets PQRSTUVWX represent the nine-digit number with the given property.
Deduction 1 : Since all numbers having even digits have to be divisible by even numbers (i.e., PQ is divisible by 2, PQRS is divisible by 4, PQRSTU is divisible by 6 and PQRSTUVW is divisible by 8), Q, S, U & W must be the 4 even numbers viz. 2, 4, 6 and 8 in some random order. Consequently, P, R, T, V & X (the odd placed digits) must be sharing the odd digits 1,3, 5,7 and 9 in some order. Deduction 2: Since PQRST is divisible by 5, T must be equal to 5. Thus, the number must be PQRS5UVWX.
Deduction 3: PQR is divisible by 3 and PQRSTU is divisible by 6, hence STU must be divisible by 3, i.e., S + T+U=S+5+U must be divisible by 3. Also, S and U are even numbers. Through trial and error you should realise that there are only two ways this could happen:
(a) If STU represents 258
or (b) If STU represents 654
Note: S can only be 2 or 6 since for PQRS to be divisible by 4, RS should be divisible by 4 and R being an odd number, S can only be 2 or 6.
Deduction 4: The following 4 options of filling in the other two even numbers (4 and 6 in case STU is 258 or 2 and 8 in case STU is 654) emerge:
$\begin{array}{cccccccccc}& P& Q& R& S& T& U& V& W& X\\ Possibility1& & 4& & 2& 5& 8& & 6& \\ Possibility2& & 6& & 2& 5& 8& & 4& \\ Possibility3& & 2& & 6& 5& 4& & 8& \\ Possibility4& & 8& & 6& 5& 4& & 2& \end{array}$
Digit sum = 4 + 5 = 9. Hence (a).