Question

a three digit number is equal to 17times the sum of its digits. if 198 is added to the number, the digits are interchanged the addition of first and third digit is 1 less than middle number. find the number.

Answer 1

let the hundreth tenth and ones digit of the given no.

then the no. will be

100x+10y+z

Given: the number is 17 times the sum of digits

=>100x+10y+z=17 (x+y+z)=17x+17y+17

=>100x-17x+10y-17y+z-17z=0

=>83x-7y-16z=0 ...1

also if 198 is added to the number the digits interchanged

=>100+10y+z+198=100z+10y+x

=>100x-x+10y-10y+z-100z+198=0

=>99x-99z+198=0

=>x-z+2=0

=>z=x+2 .....2

and the sum of the first digit is less the middle digit by unit

=>x+z=y-1

=x-y+z+1=0 ....3

putting value of z from 2 in 1 we get

83x-7y-16 (x+2)=0

=>83x-7y-16x-32=0

=>67x-7y=32 ...4

puttin value of z from 2 in 3 we get

x-y+x+2+1=0

=>2x-y+3=0

=>y=2x+3 ....5


putting value of y from 5 in 4 we get

67x-7 (2x+3)=32

67x-14x-21=53

=>x=1 .....6

putting the value of x from 6 in 5 we get

y=2+3=5 ..7

putting value of x and y from 6 and 7 in 3 we get

1-5+z+1=0

z=3


hence required no.is 153