Question

a two digit number is obtained by either multiplying sum of digits by 8 and adding one or by multiplying the difference of digits by 13 and adding 2. find the number.
Is the answer 14 or 41 correct?

Answer 1

example if number is 54, it can be written as 50+4 and 5*10+4
so there is one digit in 10s place and a digit in ones place.
let the digit in tens place be x and ones place be y
then the number will be 10x+y as shown in the example above.

given that the number is also obtained by 8(x+y)+1 and by 13(x-y)+2 ( here x-y and y-x both possiblitues must be considered)
we get 14 when we take y-x and 41 when we take x-y
so both 14 and 41 is the reqd answer. answering one of them would be insufficient and both should be written

Answer 2

Let the digits be x and y
10x+y=8(x+y)+1
10x+y=8x+8y
10x-8x+y-8y=1
2x-7y=1   (1)

10x+y=13(x-y)+2
10x+y=13x-13y+2
10x-13x+y+13y=2
-3x+14y=2   (2)
 
(1)×2:- 4x-14y=2   (3)

Adding (2) and (3),we get:-
-3x+4x+14y-14y=2+2
x=4

Substituting x in (1),
2(4)-7y=1
-7y=1-8
y=-7/-7
y=1

The number can be either 14 or 41 as there are two cases on difference of digits(x-y=0 and y-x=o)