Question

A two-digit number is such that the product of its digits is 20. If 9 is added to the number, the digits interchange their places. Find the number.

Answer 1

Let the x be th number at units place and y be the number at digits place
number will be =x+10y
Reverse number will be 10x+y
Product of digit is 20
Therefore,xy=20 -----{1}
original number + 9=reverse number
x+10y +9=10x +y
9x-9y=9
x-y=1-------{2}
using formula (x+y)²=(x-y)²+4xy
(x+y)²=(1)²+4(20) [ by using equation 1 and 2]
(x+y)²=1+80=81
(x+y)=9-----(3)
solve equation 3 and 2
x+y=9
x-y=1
_______
by adding, we get,
2x=10
⇒x=5
y=9-5
y=4
so x and y are 5 and 4
so number is x +10y =5 +10x4=5+40=45

∴Required number is 45