Question

eight times a two digit number is equal to three times the number obtained by reversing the order of its digits .if the difference between the digit is 5 ,find the number.

Answer 1

let the number be  A B          its value is  10 A + B
Reverse of the number BA      it s value is  10 B  + A

given  8 * (10A +B) = 3 (10 B + A )
80 A + 8 B = 30 B + 3 A
77 A - 22 B =  0
7A - 2 B = 0            =>  2B =  7A

we know that B is between  0 and 9.    A is between 0 and 9
B and A are not zeros.

A has a factor 2  and B has a factor 7 for the above equation to be true. There is no other way.
There is only one such possibility :  A = 2  and  B = 7

The number is  27.     
To solve this we do not need the second data, that is, difference between the digits is 5.  This information is not needed.

Answer 2

$<b>Let one's digit=y$

$<b>Units digit=x$

$<b>Number$=10x+y

⇒8(10x+y)=3(10y+x)

⇒80x+8y=30y+3x

⇒77x=22y

⇒7x=2y−−−(1)

Also, x−y=5−−−(2)

$<b>from eq(1) and (2)$

⇒x−y=5

⇒x+72y−y=5

⇒2y−7y=35

⇒−5y=35

⇒y=−7

⇒x−y=5

⇒x+y=5

⇒x=−2

$<b>Number=−27$

or

⇒y−x=5       

⇒y−72y=5

⇒5y=35

⇒y=7

⇒7−x=5

⇒x=2

$<b>Number=27$