When a two-digit numberis multiplied by the sum of its digits, the product is 424. When the number obtained by interchanging its digits is multiplied by the sum of the digits, the result is 280. The sum ofthe digits of the given number is:

Let the ten's digit x and unit digit be y.
Number = 10x + y
A two-digit number is multiplied by the sum of its digits, the product is 424..
So,
(10x + y)(x + y) = 424
$$10x^2 + 11xy + y^2 = 424$$ ---(1)
And,
(10y + x)(x + y) = 280
$$11xy + 10y^2 + x^2 = 280$$ ---(2)
Eq (1) $$\div$$ eq (2),
$$\frac{10x + y}{10y + x} = \frac{424}{280}$$
$$\frac{10x + y}{10y + x} = \frac{53}{35}$$
10x + y = 53 ---(3)
10y + x = 35 ---(4)
Eq (4) multiply by 10,
100y + 10x = 350 ---(5)
From eq (3) and (5),
99y = 297
y = 3
From eq(4),
30 + x = 35
x = 5
Number = 10x + y = 10 $$\times$$ 5 + 3 = 53
The sum of the digits of the given number = 5 + 3 = 8