Find-all-the-integers-which-are-equal-to-11-times-the-sum-of-their-digits-

Question Number 18385 by Tinkutara last updated on 19/Jul/17

Find all the integers which are equal to

11 times the sum of their digits .

Answered by mrW1 last updated on 19/Jul/17

{let}^{'} s see if such a number can have only

one digit :

sum of digits ⩾ 1

11 times sum of digits ⩾ 11

but a number with one digit is ⩽ 9

\Rightarrow the requested number must have

at least 2 digits .

{let}^{'} s see if such a number can have 4

or more digits :

sum of digits ⩽ 4 \times 9 = 36

11 times sum of digits ⩽ 11 \times 36 = 396

but a number with 4 digits is ⩾ 1000

\Rightarrow the requested number must have

less than 4 digits .

case 1 : number with 2 digits

{let}^{'} s say it is ab

10 a + b = 11 (a + b)

10 b + a = 0

\Rightarrow no solution

case 2 : number with 3 digits

{let}^{'} s say it is abc

100 a + 10 b + c = 11 (a + b + c)

89 a = 10 c + b

the only solution is :

a = 1

c = 8

b = 9

\Rightarrow there is only one such number

it is 198

Commented by Tinkutara last updated on 20/Jul/17

Thanks Sir!