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

1
by lomateps

2014-08-17T04:40:29+05:30

### This Is a Certified Answer

Certified answers contain reliable, trustworthy information vouched for by a hand-picked team of experts. Brainly has millions of high quality answers, all of them carefully moderated by our most trusted community members, but certified answers are the finest of the finest.
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