Question

Find all 3 digit natural numbers which are 12 times as large as sum of their digits

Answer 1

let us say the 3 digit natural number is A B C    (like 674).  A,B,C are digits and A is not 0.

Then  Value of  ABC  =  12 * (A+B+C)

     100 * A + 10 * B + C = 12 * (A+B+C)    = 12 A + 12 B + 12 C

     88 A  = 2 B  + 11 C            So 2 B = 11 (8A - C)          Means 8A - C is even.
That means C is even as 8A is even.  Choose numbers with C as even: 0,2,4,6,8

A >= 1  That means 88A >= 88.  But 2B <=18.  That means 11C > 70.
That means C >= 7.  So C = 8 only.

    88A = 2 B + 88           44A = B + 44        B = 44(A-1)
   This is possible only if  A = 1 and B=0.       For A>1, B is > 9 which is not possible.

ONLY ONE SOLUTION      108