Question

A three digit number is equal to 17 times the sum of the digits. If the digits are reversed the new number is 198 more than the original number. The sum of the extreme digits is less than the middle digit. Find the original number.

Answer 1

The original number for the given statement is 153

Step-by-step explanation:

Given as :

Statement I

A three digit number is equal to 17 times the sum of the digits.

Let ones digit = 1 × x  = x

tens digit = 10 × y = 10 y

hundreds digit = 100 × z = 100 z

So, original number = 100 z + 10 y + x

According to question

The sum of digits = x + y + z

So,

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

Or, 100 z + 10 y + x = 17 x + 17 y + 17 z

Or, ( 100 z - 17 z )  + ( 10 y - 17 y ) + ( x - 17 x ) = 0

i.e,  83 z - 7 y - 16 x = 0                                     -----------------------1

Again

Statement II

If the digits are reversed the new number is 198 more than the original number

So, The reverse digits = 100 x + 10 y + z

i.e   100 x + 10 y + z  = 100 z + 10 y + x + 198  

Or,  ( 100 x - x ) + ( 10 y - 10 y ) + ( z - 100 z ) = 198

Or,   99 x + 0 - 99 z = 198

Or,   x - z = $\dfrac{198}{99}$

O,   x - z =  2                    ---------------------(2)

So,    z = x - 2                                 

Again

Statement III

The sum of the extreme digits is one less than the middle digit.

i.e z + x = y - 1                                            .......................3

Now, Solving eq 2 and 3                                      

 ( x - z )  + ( z + x ) = 2 + y - 1

Or, 2 x - y = 1

So, y = 2 x - 1                                                .................4

Now, Put the value of y and z in eq 1

∵ 83 z - 7 y - 16 x = 0

Or, 83 ( x - 2 ) -7 ( 2 x - 1 ) - 16 x = 0

Or,  83 x - 166 - 14 x + 7 - 16 x = 0

Or, 53 x = 159

∴          x =  $\dfrac{159}{53}$

i.e         x = 3

Now, put the value of x in eq 4

y = 2 × 3 - 1  

i.e y = 5

Now, Put the value of x in eq 2

z = 3 - 2  

i.e  z = 1

Putting the value of x , y , z in original number we get

So, The original number =  100 z + 10 y + x

Or, The original number =  100 × 1 + 10 × 5  + 3

Or, The original number = 100 + 50 + 3

∴ The original number = 153

Hence, The original number for the given statement is 153  Answer